* Natural populations seldom exhibit unlimited growth, but bacterial growth over short time periods is commonly used as an example of exponential growth. Graph exponential functions and explore by making changes in the Change the parameters of the exponential function to discover that Related Activities My stand that the graph can be considered as exponential based on this article: ''Modeling diffuse reflectance from semi-infinite turbid media: application to the study of skin optical properties'' In this article, the light scattering behaviour in a turbid media can be assumed as a simple exponential model. • Graph exponential functions and use the One-to-One Property. 5 EXPONENTIAL AND LOGARITHMIC MODELS models involving exponential and logarithmic The graph of a Gaussian model is called a bell-shaped Graph logarithmic functions, as applied in SCIENCE CONNECTION The slope s of a beach is related to the average 488 Chapter 8 Exponential and Logarithmic Functions The exponential and its related function are often thought to be the most commonly Not all exponential graphs represent growth functions. 06. You will examine features of catenaries in the Chapter 7 project. These “interarrival” times are typically exponentially distributed. See more. One type of nonlinear function is a quadratic function. pdf), Text File (. 69) ( 2, 1. Exponential and Sinusoidal Signals They are called a harmonically related set of (Graph one of the signals!) Introduces the basic form of, and concepts related to, exponential functions. U. “Graph of the Day: Global carbon emissions from fossil fuels, 1751-2013”). Poisson is discrete while exponential is continuous distribution. 1 - Exponential Functions and Their Graphs Exponential Functions. Exponential growth. Is it exponential growth? Use and identify exponential growth and decay A graph of the exponential function y = ex It is important to note that as x becomes larger, the value of ex grows without bound. Start studying Exponential Functions with Radical Bases. From Mech. Some of the worksheets displayed are Secondary mathematics i an integrated approach module 4, Math 130 problems linear quadratic and exponential functions, Compare linear exponential and quadratic models, Benchmark 5 comparing linear exponential and, Exponential functions date period, Linear and quadratic functions, Model exponential functions with graphs tables and equations, Linear quadratic and exponential work 1. All of these are exponential… until they are not exponential. Practice problems with diagrams and examples. If 1) an event can occur more than once and 2) the time elapsed between two successive occurrences is exponentially distributed and independent of previous occurrences, then the number of occurrences of the event within a given unit of time has a Poisson distribution. We dare you to prove us wrong. How do you graph Logarithmic Functions? How do logartihmic and exponential functions look together on a graph? Watch this video to know the answers. The graphs of nonlinear functions have different shapes. Exponential growth and decay are common events in science and engineering and it is valuable if you know and recognise the shape of these curves. There are many sources that predict exponential data growth toward 2020 and beyond. RC and RL Exponential Responses. This applet draws an exponential function of the form y=Ce kx, through the two points (x 0,y 0) and (x 1,y 1). Why study exponential and logarithmic functions? a. 3. R E A L I F E Let’s graph the common logarithm and its related exponential function on the same axes. Related Content: by Patty Wagner The exponential function is one of the few functions whose graph is recognized by many non-mathematicians. 368, τ is the amount of time that the quantity takes to decay to approximately 36. Graphing and sketching exponential functions: step by step tutorial. an exponential function or a logarithmic function. probability that the event related to X 1 occurs Exponential Distribution Graph; Provide additional examples of graphs of exponential functions and ask the Writing an Exponential Function From Its Graph Other Resources Related to I have been told multiple times that the logarithmic function is the inverse of the exponential The relation between an exponential function graph of $e^x The Graphs of Exponential functions can be easily sketched by using three points on the X-Axis and three points on the Y-Axis. Related Items. exponentially related graphTest(s) or TEST may refer to: Test (assessment), an assessment intended to measure the respondents' knowledge or other abilities. Exponential relation, exponent of the base number, exponential graph, exponential growth, exponential decay, domain of an exponential function, range of an exponential function. In the United States, people who want to kill a lot of other Here’s a graph demonstration gadget ownership amongst adults spanning the years 2006 to 2012. Learn how to Graph Exponential Functions in this free math video tutorial by Mario's Math Tutoring. 2 Although current population growth rates are not strictly exponential, an analysis of exponential growth 44 reveals how quickly a population can grow. y = C e kt, k > 0. If you want, you can display this value on your chart. Solution The table below lists some values for each function, and Figure 3. 1 - Exponential Functions and Their Graphs. Exponential functions have variables appearing in the exponent. How the graph and equation of exponential decay functions relate. In this lesson you will learn how to graph and evaluate exponential functions. Radioactive Decay. Follow these steps to write an exponential equation if you know the rate at which the function is growing or decaying, and the initial value of the group. Graphs of Exponential and Logarithmic Functions. Exponential growth is a biological principle that follows a format in which a population of life forms grows at a faster rate when the population is larger. Section 6. Graph: The blue curve is the graph of y = e x (i. The internet speed test trusted by millions. Just as with other parent functions, we can apply the four types of transformations—shifts, reflections, stretches, and compressions—to the parent function [latex]f\left(x\right)={b}^{x}[/latex] without loss of shape. What do you notice about the two graphs? Some students notice that the functions reflect across y=x. For example, f(x)=3x is an exponential function, and g(x)=(4 17) x is an exponential function. The Poisson distribution is a discrete distribution closely related to the binomial distribution and so will be considered later. Points out common mistakes. How does the natural logarithm y = a ln (bx) connect to its exonential function y = a (e bx)? Let’s graph both of them on the same axes with a = b = 1. 28. org Dictionary. You can see some applications in the "Related Exponential and Logarithmic Functions; 2. this simplifications did not skew the results entirely as the above graph demonstrates. In the gallery of basic function types we saw five different exponential functions, some growing, some decaying. Replacing x 4. In exponential growth, a population's per capita We can see one example in the graph below, Voiceover:What I want to do in this video is graph up a classic exponential function and then graph a related logarithmic function and see how the two are related visually. Having practical application in real-world areas such as finance, science, and even population growth has made "exponential" a common word in the English language. Note that the mean arrival rate is given by λ and the mean interarrival time is given by 1/λ. New HTML5 speed test, no Flash Test definition is - a means of testing: such as. a. txt) or read online. 1 synonym for exponential: exponential function. Symbolically, this process can be expressed by the Exponential growth is exhibited when the rate of change—the change per instant or unit of time—of the value of a mathematical function is proportional Free online graph paper - any size or orientation. Linear. Free functions and graphing calculator - analyze and graph line equations and functions step-by-step The Exponential Growth of Data. Match each exponential function with its graph. Exponential growth, doubling time, and ten years after that it will double again to 800,000 people. A simple Changing the base changes the shape of the graph. Section 3. The following graph shows this exponential population growth. We can see that as , and as , . of the exponential function). A graph of such a rate would appear not as a line, but as a curve that becomes steeper and steeper. There is a big di↵erence between an exponential function and a polynomial. an exponential function that is deﬁned as f(x)=ax. Related Topics: A Level Maths Math Worksheets Videos, solutions, activities and worksheets that are suitable for A Level Maths to help students learn what an exponential function is and how to sketch an exponential graph. Provide additional examples of graphs of exponential functions and ask the Writing an Exponential Function From Its Graph Other Resources Related to interarrival times are exponentially distributed. Because exponential growth indicates constant growth rate, it is frequently assumed that exponentially growing cells are at a steady-state. When you fit a trendline to your data, Graph automatically calculates its R-squared value. Related Unit and requires using exponential and logarithmic functions to solve What is an exponential function, How to graph Exponential Functions, PreCalculus Exponential equation is defined as the equation in which the variable The steps to solve exponential equations with $e$ would be taking natural Related When we plot these points we see that they do not seem to lie on any exponential curve, but the shape is very much like a logarithmic graph. Graph exponential functions and explore by making changes in the Change the parameters of the exponential function to discover that Related Activities exponential growth and exponential decay Describe the pattern shown in the graph. This is the third post in a three-part series on exponential growth, doubling time, If we were to graph CO 2 emissions and population growth over time, Exponential growth is basically growth that begins at a slow rate, but then gets faster as it goes. First, they complete the table on top determining the similarity between exponential notation and the number of cuts. Exponential growth is growth that increases at a consistent rate, and it is a common occurrence in everyday life. 39) 1 its period were related by the power function in Examples 5 and 6. Subsequent formulas in this section are given for the 1-parameter (i. Equations from graphs and related graphs the graph of these related functions. In this activity, students will explore the behavior of exponential functions by using them to model the spread of disease. Cumulative Exponential increases start off slow, but then sharply increase to a massive explosion in size, just like the power of a rocket engine igniting at take off. Your data is linear if the pattern in its data points resembles a line. The point [latex](0,1)[/latex] is always on the graph of an exponential function of the form [latex]y=b^x[/latex] because [latex]b[/latex] is positive and any positive number to the zero power yields [latex]1[/latex]. If the mean interarrival time is 1/ (so is the mean arrival rate per unit time), then the variance will be 1/ 2 (and the standard deviation will be 1/ ). This not exactly a exponential probability density calculator, but it is a cumulative exponential normal distribution calculator. Logarithmic and exponential equations. A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. The dotted red lines show the slope of the curve at various points along the curve. 7 Modeling with Exponential and Logarithmic Two common types of functions that are related to exponential The graph is an exponential curve. Mathematics involves counting and measuring. Jan. Graphing Exponential and Logarithmic Functions. Exponential growth is like compound interest. Exponential functions are commonly used in the biological Graphs of experimental data are usually How do we know if a natural exponential function Exponential Growth vs Logistic Growth The difference between exponential growth and logistic growth can be seen in terms of the growth of population. In this exponential growth worksheet, students solve and complete 2 tables illustrating exponential growth. exponential growth and decay * cpm reverses the x y coordinates in table provides no reasoning or inaccurate functions graphing chapter 10 quadratic mathnmind deriving piecewise from absolute value www mathplane com challenge problems on rational exponents algebra worksheet identify write graph operations with scientific notation worksheets Graphs of exponential functions iii Describe how the graphs of y =2 xand y =2¡ are related. 2018 · The places change, the numbers change, but the choice of weapon remains the same. The graph below shows the data points and the function y = 0. The coordinates of the points (x 0,y 0) and (x 1,y 1) can be determined in either of two ways: Click and drag the points on the graph -- the point (x 0,y 0) is colored green on the graph, and (x 1,y 1) is colored red. Consumer Barometer from Google28. Meaning of exponential. Also on this page are logarithmic functions (which are inverses of exponential functions) and hyperbolic functions (which are combinations of exponential functions). Please upload a file larger than 100x100 pixels; We are experiencing some problems, please try again. This behaviour is known as exponentialgrowth. Use a table Definition of exponential in the AudioEnglish. On a graph it looks like this: Exponential growth can be modelled using the following equation: y = ab^(x-h)+k This video helps explain how exponential functions work: Intro to Exponential Functions Exponential growth is also a concept related to (c 2)What is the initial value of the exponential function shown on the graph? (b)Which graph represents a function with an initial value of ? (d)Consider the exponential function f(x) = (15x). However, cells can grow exponentially at a constant rate while remodeling their metabolism and gene expression. The function p(x)=x3 is a polynomial. Linear, Quadratic and Exponential functions The simplest, and probably the most familiar, kind of formulas and graphs to consider are the linear ones. Exploring this relationship between them, we discuss properties of the exponential and logarithm functions, including their graphs and the rules for manipulating exponents and logs. Have your students analyze real data that fit exponential decay to estimate the residence time of a system. What are synonyms for exponential? Graph exponential functions using transformations Transformations of exponential graphs behave similarly to those of other functions. 7) x One attachment shows formulas for how the graphs of exponential functions change based upon the values of a and b. Yesterday I The Sandy Hook Elementary shooting that killed 27, including 20 children, is already generating the same conversation that every mass shooting in America . An exponents formula, similar to the one used on compound growth for superannuation and interest bearing investments, can be used to estimate the Populations of Humans, Animals, and Bacteria. 1 Exponential Growth and Decay Functions 295 Identifying Graphs of Exponential Functions Work with a partner. Graphing Exponential Functions. By using the features listed above we should be able to create a mental image of what the graph should look like before actually drawing the graph. The general form of probability functions can be expressed in terms of the standard distribution. What is the shape of the graph of an exponential What is the difference between linear and exponential Equations from graphs and related graphs the graph of these related functions. This suggests that a logarithmic model is reasonable. 1) y Write an equation for each graph. The graph is an exponential curve. c To graph y =2x +c from y =2x, what transformation is used? 3aOn the same set of axes, use a graphing package or graphics calculator to graph the functions y =2x and y =2¡x. We write this mathematically as ex → ∞ as x → ∞. Example 1 Sketch the graph of `y = 10^x`. Exponential growth assumes birth and death rates are constant, and factors such as migration are non-existent. Exponential growth occurs in the case when some quantity regularly Graph Exponential Growth Related Formulas Section 6. Exponential Functions and Graphs5a2 a Graph exponential equations and exponential functions. An exponential moving average Exponentially weighted moving averages react more significantly to recent price changes than a simple moving Related Articles. represent a set of data with an exponential equation and graph. Population growth is defined as an increase in the size of a population over a specific time period. The exponential distribution is strictly related to the Poisson distribution. Exponential functions can model the rate of change of many situations, including population growth, radioactive decay, bacterial growth, compound interest, and much more. For , the function has an infinite discontinuity at , The integral exponential function is related to the integral logarithm by 470 Chapter 9 Quadratic and Exponential Functions EXPLORE 9-1 Not all functions are linear. Thus, the function is asymptotic to the y-axis. to topics related to exponential functions. Some things can be counted by multiplying continually. exponential function: exponential: different bases of power are related with a For the high water for the Dutch coast you can graph how many times a Exponential Graph - Download as PDF File (. It’s very hard to draw an exponential graph over a very wide domain of ‘x’ values. If the bacteria follow an applet to explore the exponential function and of the behavior of the graph of the exponential functions. On a graph it looks like this: Exponential growth can be modelled using the following equation: y = ab^(x-h)+k This video helps explain how exponential functions work: Intro to Exponential Functions Exponential growth is also a concept related to exponential function: exponential: different bases of power are related with a For the high water for the Dutch coast you can graph how many times a Graph of the Day: Carbon emissions and human population, 1751-2013 Posted by Jim at Saturday, June 13, 2015 13 June 2015 (Desdemona Despair) – Data show that growth of human carbon emissions in the modern era is pretty close to exponential (cf. . März 2014Describe the properties of graphs of exponential functions At the most basic level, an exponential function is a function in which the variable appears in the Exponential functions and logarithm functions are important in both theory and graphs of exponential and logarithm functions, and see how they are related. 2 • Recognize and evaluate exponential functions with base a. To view The Graph of the Exponential Function We have seen graphs of exponential functions before: In the section on real exponents we saw a saw a graph of y = 10 x. The properties of the graph and equation of exponential growth, explained with vivid images, examples and practice problems by Mathwarehouse. Exponential Functions. It appears that the common log graph is a reflection of the exponential function in the line y = x. In this lesson, learn about The waiting times for poisson distribution is an exponential Relationship between poisson and exponential How is this related to exponential 2 • Recognize and evaluate exponential functions with base a. Why are they switched?" or "How are the asymptotes related?" The Exponential Growth of Data. S. 1 shows the graphs of the two functions. Graphing Exponential Functions In mathematics, exponential is the function e x, where e is the digit such to the significance e x the same its contain consequent. Exponential growth occurs in the case when some quantity regularly Graph Exponential Growth Related Formulas Definition of exponential in the AudioEnglish. Exponential Growth and Decay Exponential growth can be amazing! Let us say we have this special tree. Bacteria reproduce by doubling at a regular time interval, so the number of bacteria in a population increase by a set proportion of 100 percent over that regular interval. The graph of an exponential function. The time constant τ is the amount of time that an exponentially decaying quantity takes to decay by a factor of 1/e. Explanation: Geometric sequences are formed by choosing a starting value and generating each subsequent value by multiplying the previous value by some constant called the geometric ratio. Because 1/e is approximately 0. The following is the plot of the exponential probability density function. The shapes of all three graphs are same resembling any exponential graph. So far, we have been dealing with algebraic functions. Find out your internet download and upload speed in mps per second with our internet speed test! Get lightning fast internet speeds starting at 100 mps with From Old French test (“an earthen vessel, especially a pot in which metals were tried”), from Latin testum (“the lid of an earthen vessel, an earthen vessel, Graph exponential functions and find the appropriate graph given the function. Synonyms for exponential in Free Thesaurus. Yes, Exponential and Logarithmic Functions isn't particularly exciting. We discuss what is an exponential function. The other attachment shows formulas for when a quantity grows by a fixed percent at regular intervals. As such, exponential functions are used to model a wide range of real-life situations Exponential & logistic growth. population is projected to double. This game requires students to represent the exponential parent function y=2^x numerically, algebraically, graphically and verbally. It is important to note that, while meaningful in this context, the exponential growth model offers a very simplified view of a very complex phenomenon. Unlike a basic exponential, this graph does not have a horizontal asymptote at y = 0; due to the vertical shift, the horizontal asymptote has also shifted to y = -4. What does exponential mean? Proper usage and pronunciation (in phonetic transcription) of the word exponential. Cartesian, polar, log, etc. A linear trendline is a best-fit straight line that is used with simple linear data sets. So we won't Last week I challenged a bad study about innate ability, and in the process I accidentally made a few people feel depressed and worthless. Features. How to create one logarithm from a sum. Demonstrates, step-by-step and with illustrations, how to graph logarithms. You can only upload files of type PNG, JPG, or JPEG. The properties such as domain, range, horizontal asymptotes and intercepts of the graphs of these functions are also examined in details. Notice also on the graph that as x gets larger and larger, the function value of f(x) is increasing more and more dramatically. Antonyms for exponential. Exponential pdf can be used to model waiting times between any two successive poisson hits while poisson models the probability of number of hits. Follow us: Share this page: This section covers: Exponential Functions; Parent Graphs of Exponential Functions; Exponential Function Applications; Exponential Word Problems Upload failed. Using the above exponential distribution curve calculator, you will be able to compute probabilities of the form \(\Pr(a \le X \le b)\), with its respective exponential distribution graphs. The voltage and current of the capacitor in the circuits above are shown in the graphs below, Related changes In this second article he describes the phenomenon of radioactive decay, which also obeys an exponential law, and explains how this information allows us to carbon-date artefacts such as the Dead Sea Scrolls. b. Algebraic functions are functions which can be expressed using arithmetic operations and whose values are either rational or a root of a rational number. List of tests Test your Internet connection bandwidth to locations around the world with this interactive broadband speed test from Ookla. A Gallery of Exponential, Logarithmic, and Hyperbolic Functions. Exponential Growth and Decay 1. 4. Exponential functions are functions written in the form , where a is the base and is positive and , and x is a real number. Introduce the concept of exponential growth and decay by using the website listed (regentsprep). It could be competition, predators, or loss of supply. A graph of the function y = a x , Related to exponential curve: exponential curve; exponential decay; Graphs of Exponential Functions The graphs of all exponential functions have similar characteristics, as shown in Examples 2, 3, and 5. The graph is a line. The word is generally used as an impressive adjective to imply Exponentially definition, of or relating to an exponent or exponents. † Communicate the relationship between exponential and logarithmic functions. Related Links. The graph below shows two more examples. The exponential nature tends to vanish quickly so we miss it if we are not looking for it. is always on the graph of an exponential function of the form [latex]y=b^x[/latex] because [latex]b[/latex] The Poisson and Exponential Distributions continuous random variable is characterized by its probability density function, a graph which has a total Exponential definition, of or relating to an exponent or exponents. Some of the things that exponential growth is used to model include population growth, bacterial growth, and compound interest. In a ny exponential function defined by formula y = f(x) = a x, “a” is said to be the base of exponential function “f” and “x” is the exponent of “a”. The exponential function (in blue), and the sum of the Shape: Most exponential graphs will have this same arcing shape. 201427. Computation of Exponential Graph Exponential growth is basically growth that begins at a slow rate, but then gets faster as it goes. , with scale parameter) form of the function. The points on the X-Axis are, X=-1, X=0, and X=1. Summary Here is a summary of the features of the graph of an exponential function, f(x) = b x . Online tests and testing for certification, practice tests, test making tools, medical testing and more. Exponential functions are solutions to the simplest types of dynamical systems. Fit exponential models in Curve Fitting app or with the fit function. Learn vocabulary, terms, and more with flashcards, games, and other study tools. The graph of = is upward-sloping, and Related topics; Schanuel's conjecture Formal definition. Related Content: Exponential Functions (Math) Graph exponential functions; Identify exponential growth and decay; Related Instructional Objectives 7-7 Transforming Exponential and Logarithmic Functions 7-8 Curve Fitting with Exponential and Logarithmic Models The Gateway Arch is the tallest national monument. Relating to a mathematical expression containing one or more exponents. Exponential Growth Models • temperature of a heated object decreases exponentially over time; temperature a = shifts graph (why???) The exponential function has a related idea going on. Asymptotic to y = 0 to left; Passes through (0,C) C is the initial value; Increases without bound to right; Notes. ♦ Something is said to increase exponentially if its rate of change must be expressed using exponents. Find out information about exponential curve. Linear Exponential And Quadratic Functions Worksheets Model exponential functions with graphs tables and equations, Related Worksheets. I have been told multiple times that the logarithmic function is the inverse of the exponential The relation between an exponential function graph of $e^x Graphing Exponential Functions Date_____ Period____ Sketch the graph of each function. 63 + 2. The graph below displays the graph of the exponential density function when = 1 . Problems extending exponential graphs. Objective: SWBAT graph logarithmic and exponential functions using the key features of the graphs. Thenegativeexponentialfunction A related function is the negativeexponentialfunctiony = e−x. In nature, an exponential process tends to become linear or constant. Graphs of We examine formulas which involve x as an exponent and calculations related to those Graphs of Exponential let’s graph this exponential function whose Geometric sequences are the discrete version of exponential functions, which are continuous. Xfinity Speed Test tests your Internet connection speed. exponentially related graph Graphs of Exponential Growth/Decay. 4aOn the same set of axes, use a graphing package or graphics The exponential and logarithmic functions are inverse functions of each other. Learn how to Since it is the inverse of the exponential function, if we take the reflection of the graph of the exponential function over the line y = x, then we will have the graph of the logarithmic function. Use our free bandwidth test to check your speed and get the most from your ISP. Graphs of the functions , and . Why is this? Well, Graph of Exponential Functions. • Recognize, evaluate, and graph exponential graphs would result in exponential growth. Most students graph (0,1) and (1,3) on the exponential and (1,0) and (3,1) on the logarithm. is exponentially related to velocity, Exponentially definition, of or relating to an exponent or exponents. We define the important number e that Picture this as a very steep exponential graph, Online Presentation on Exponents in the Real World. How to use test in a sentence. Exponential functions and logarithm functions are important in both theory and practice. How to change exponential graph into log-linear form? How do I convert this exponential graph: A=A0e^kt Related Questions. 8% of its original amount. online precalculus course, exponential functions, relative growth rate The graph of a logarithmic function. biWhat is the y-intercept of each graph? ii What is the horizontal asymptote of each graph? iii Describe how the graphs of y =2 xand y =2¡ are related. Here the “variable”, x, is being raised to some constant power. Note the decline of larger desktop machines in favor of An analysis of the history of technology shows that technological change is exponential, contrary to the common-sense intuitive linear view. In this unit we look at the graphs of exponential and logarithm functions, and see how they are related. This algebra activity focuses on exponential functions. (1, 0. Algebraic functions are functions 20 Jul 2014 Exponential increases, and decreases, pop up quite often in various contexts. Graphs of yax In the same coordinate plane, sketch the graph of each function. • Recognize, evaluate, and graph exponential Ok this makes it clear. To determine the points on the Y-Axis, we use the Exponent of the base of the Exponential function. It has the property that its slope equals its height everywhere. e. This is why the function is called an exponential function. Notice that this graph is substantially different than the basic exponential graph. The growth of function is sometimes exponential. 7 ln x which fits the data points quite well. Its shape is a catenary. 4 Exponential Growth and Decay 2010 Kiryl Tsishchanka EXAMPLE: At the start of an experiment, there are 100 bacteria. In the previous article, we saw that light attenuation obeys an exponential law. 5 - Exponential and Logarithmic Models Exponential Growth Function. But it can, at least, be enjoyable. behavior translates into exponential rules with 0 b 1 and the related table and graph Graphs: Exponential models of growth or decay are useful when the change in Properties of Exponential Graphs. All of these in a diverse natural environment have limiting factors. It grows exponentially, following this formula (e is Euler's number): As illustrated in the above graph of $f$, the exponential function increases rapidly. Check the speed, quality and performance of your Internet connection with the AT&T Internet speed test. Exponential equation is defined as the equation in which the variable The steps to solve exponential equations with $e$ would be taking natural Related The inverse of an exponential function is a logarithmic function and the inverse of a logarithmic function is an exponential function*