The exponential function is y = (1/4)(4) x. A more complicated example showing how to write an exponential function. Example 1 In the previous examples, we were given an exponential function, which we then evaluated for a given input. The most commonly used exponential function base is the transcendental number e, which is approximately equal to 2.71828. Other examples of exponential functions include: $$ y=3^x $$ $$ f(x)=4.5^x $$ $$ y=2^{x+1} $$ The general exponential function looks like this: \( \large y=b^x\), where the base b is any positive constant. Example 3: Find the domain and range of the function y = log ( x ) − 3 . One example models the average amount spent (to the nearest dollar) by a person at a shopping mall after x hours and is the function, fx( ) 42.2(1.56) x, domain of x > 1. Sometimes we are given information about an exponential function without knowing the function explicitly. Examples, solutions, videos, worksheets, and activities to help PreCalculus students learn about exponential and logarithmic functions. The base b could be 1, but remember that 1 to any power is just 1, so it's a particularly boring exponential function! The graph is nothing but the graph y = log ( x ) translated 3 units down. If we have an exponential function with some base b, we have the following derivative: `(d(b^u))/(dx)=b^u ln b(du)/(dx)` [These formulas are derived using first principles concepts. Scroll down the page for more examples and solutions for logarithmic and exponential functions. Transformations of exponential graphs behave similarly to those of other functions. Sometimes we are given information about an exponential function without knowing the function explicitly. Here is a set of practice problems to accompany the Exponential Functions section of the Exponential and Logarithm Functions chapter of the notes for Paul Dawkins Algebra course at Lamar University. An Example of an exponential function: Many real life situations model exponential functions. We will assume knowledge of the following well-known differentiation formulas : , where , and , where a is any positive constant not equal to 1 and is the natural (base e) logarithm of a. The following diagram gives the definition of a logarithmic function. Example #2 Find y = ab x for a graph that includes (1, 2) and (-1, 8) Use the general form of the exponential function y = ab x and substitute for x and y using (1, 2) 2 = ab 1 2 = ab Divide both sides by b to solve for a Exponential Equations – examples of problems with solutions for secondary schools and universities Finding Equations of Exponential Functions. THE INTEGRATION OF EXPONENTIAL FUNCTIONS The following problems involve the integration of exponential functions. Graph the function on a coordinate plane.Remember that when no base is shown, the base is understood to be 10 . 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