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# Exponential Functions

Recall that an exponential function is one that has the form y = bx, where b > 0 and b 1. The logarithmic function is the inverse of the exponential function.

Before we study logarithmic functions we will review some of the properties of exponential functions.

Here is an exponential function: y = 2x

To graph y = 2x we start with values of x and find the values of y. Then, we plot the ordered pairs on a Cartesian coordinate system.

For example,

when x = 0, we have y = 20 = 1;

when x = 4, we have y = 24 = 16.

Next, we will calculate more values for y = 2x.

 x f(x) = 2x (x, y) -20 1 2 4 f(-2) = (2)-2 = (1/2)2 = 1/4f(0) = (2)0 = 1 f(1) = (2)1 = 2 f(2) = (2)2 = 4 f(4) = (2)4 = 16 (-2, 1/4)(0, 1) (-1, 2) (2, 4) (4, 16)

We found the above ordered pairs by selecting values for x and then calculating y. Sometimes, however, we know a value of y and must find the value of x.

For example, suppose we know that for y = 2x, y = 8. To find the value of x we can look at the graph of y = 2x. From the graph, we can see that when y = 8 the value of x is 3.

 Hereâ€™s another way to find the value of x when y = 8.Substitute 8 for y. Then, ask yourself, â€œ2 to what power results in 8?â€. The answer is 3. y = 2x8 = 2x   8 = 23

Now, suppose you know that for y = 2x, y = 6 and you want to find the value of x. That is, you want to solve the equation 6 = 2x. We can see from the graph that x must be between 2 and 3 since 22 is 4 and 23 is 8. To solve an equation such as 6 = 2x we introduce logarithms.