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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)
-2

0

1

2

4

f(-2) = (2)-2 = (1/2)2 = 1/4

f(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.

Here’s another way to find the value of x when y = 8.

Substitute 8 for y.

Then, ask yourself, “2 to what power results in 8?”.

The answer is 3.

y = 2x

8 = 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.

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