Exponential Functions
Recall that an exponential function is one that has the form y = b^{x},
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 = 2^{x}
To graph y = 2^{x} 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 = 2^{0} = 1;
when x = 4, we have y = 2^{4} = 16.
Next, we will calculate more values for y = 2^{x}.
x 
f(x) = 2^{x} 
(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 = 2^{x}, y = 8. To find the value of
x we can look at the graph of y = 2^{x}. 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 = 2^{x} 8 = 2^{x}
8 = 2^{3} 
Now, suppose you know that for y = 2^{x}, y = 6 and you want to find the
value of x. That is, you want to solve the equation 6 = 2^{x}. We can see
from the graph that x must be between 2 and 3 since 2^{2} is 4 and 2^{3} is 8.
To solve an equation such as 6 = 2^{x} we introduce logarithms.
