3.2 Logarithmic Functions and Their Graphs

The logarithmic function f with base a is denoted as:
    f(x)=log_ax
where a>0, a≠1, and x is a real number.

The logarithmic function f(x)=log_ax is the inverse of the exponential function f(x)=a^x.

 These are the graphs for f(x)=log₁₀x (in blue) and f(x)=10^x (in red).

Notice that that the logarithmic graph is a transformation of the exponential graph.

       THE KEY TO EVERYTHING

With a logarithmic function, y=log_ax implies that a^y=x.

For example:
log₂4=2
implies that
2²=4.

When evaluating logarithmic functions, it is important to keep this in mind.

For example:
   log₅125=x
(We can solve for x by remembering The Key to Everything.)
   5^x=125
(125 is equal to 5³.)
   5^x=5³
(If 5 raised to the x power is equal to 5 raised to the 3^rd power, then...)
   x=3
(...x is equal to 3.)

       THE PROPERTIES OF LOGARITHMS

• log_a1=0 because a⁰=1
• log_aa=1 because a¹=a
• log_aa^x=x because a^x=a^x
• If log_ax=log_ay, then x=y.

The natural logarithmic function, similar to the natural exponential function, uses the number e as a base.
   For example: f(x)=log_ex
The natural logarithmic function is also rewritten in this way:
f(x)=ln x where ln denotes log_e.

The common logarithmic function is a logarithm with the base 10.
   For example: f(x)=log₁₀x
The common logarithmic function is also rewritten in this way:
f(x)=log x where log denotes log₁₀.

       GRAPHS OF LOGARITHMIC FUNCTIONS

Recall that the graph of the logarithmic function f(x)=log_ax is the inverse of the exponential function f(x)=a^x.
The graph of a logarithmic function is its corresponding exponential graph reflected across the line y=x.

These are the graphs for f(x)=log₁₀x (in blue), f(x)=10^x (in red), and f(x)=x (in orange).

In order to graph a logarithmic function, you should first graph its corresponding exponential function. Because they are inverses, you can find the corresponding exponential function by switching the x and y values.



Additionally, because they are inverses, the domain of one function is the range of the other, and the range of one function is the domain of the other.

   For example:
The domain (the possible x-values) of the function f(x)=log x is (0, ∞).
Therefore, the range (the possible y-values) of the function f(x)=10^x is (0, ∞).
The domain of the function f(x)=10^x is (-∞, ∞).
Therefore, the range of the function f(x)=log x is (-∞, ∞).