Logarithmic Functions
The statement 8 = 2^{3} is in exponential form with base 2.
An equivalent way of writing 2^{3} = 8 is the logarithmic form:
We read log_{2}8 = 3 as, â€œthe logarithm to the base 2 of 8 is 3.â€ Or, more
briefly, â€œthe log base 2 of 8 is 3.â€
The statements 8 = 2^{3} and log_{2}8 = 3 are two ways of expressing the same
relationship between 8 and the cube of 2.
So what is a logarithm? A logarithm is an exponent. We can see this by
noting the following: In the equation 8 = 2^{3} we know 3 is the exponent.If we rewrite 8
= 2^{3} in logarithmic form we have log_{2}8 = 3.
So the logarithm, log_{2}8, is equivalent to 3, an exponent.
Definition â€” Logarithmic Function
A logarithmic function is a function that has the form:
f(x) = log_{b}x
where b is a real number, b > 0, and b ≠ 1;
x is a real number, x > 0.
The domain is all positive real numbers.
The range is all real numbers.
The inverse of an exponential function is a logarithmic function. This
means that
â€¢ If f(x) = b^{x} then f^{1}(x) = log_{b}x. Both functions have base b.
â€¢ The domain of one function is the range of the other.
â€¢ Their graphs are mirror images of each other about the line y = x.
Since they are inverses, we can write an exponential equation as a log
equation and vice versa.
Note:
The inverse of f(x) = 2^{x} is
f^{1}(x) = log_{2}x.
The inverse of f(x) = log_{2}x is
f^{1}(x) = 2^{x}.
Definition â€” Exponential and Logarithmic Forms
Exponential form If b^{L} = x 
then 
Logarithmic form log_{b}x = L 
Example 1
Rewrite in logarithmic form:
a. 5^{x} = 625
b. 3^{4} = x
c. b^{2} = 100
Solution
The statements are given in exponential form, b^{L} = x.
Rewrite them in the equivalent logarithmic form, log_{b}x = L.
a. The base is 5. The exponent is x.
b. The base is 3. The exponent is 4.
c. The base is b. The exponent is 2. 
log_{5}625 = x
log_{3}x = 4
log_{b}100 = 2 
Example 2
Rewrite in exponential form:
a. log_{6}x = 2
b. log_{4}64 = x
c.
Solution
The statements are given in logarithmic form, log_{b}x = L.
Rewrite them in the equivalent exponential form, b^{L} = x.
a. The base is 6. The exponent is 2.
b. The base is 4. The exponent is x.
c. The base is 5. The exponent is 2. 
6^{2} = x
4^{x} = 64

Note:
Remember,
