Property 1: because
.
Example 2: In the equation
, the base is
and the exponent is
0. Remember that a logarithm is an exponent, and the corresponding
logarithmic equation is
.
Example 3: Use the exponential equation
to write a logarithmic
equation. The base x is greater than 0 and the exponent is 0.
The corresponding logarithmic equation is
.
Property 2: because
.
Example 5: In the equation
, the base is 87, the exponent is 1,
and the answer is 87. Remember that a logarithm is an exponent,
and the corresponding logarithmic equation is
.
Example 6: Use the exponential equation
to write a logarithmic
equation. If the base p is greater than 0, then
.
Property 3: because
.
Example 7: Since you know that
, you can write the logarithmic
equation with base 3 as
.
Example 8: Since you know that
, you can write the logarithmic
equation with base 13 as
.
Example 9: Use the exponential equation
to write a logarithmic
equation with base 4. You can convert the exponential equation
to the logarithmic equation
. Since the 16 can be written as
, the equation
can be written
.
The above rules are the same for all positive bases. The most common bases are the base 10 and the base e. Logarithms with a base 10 are called common logarithms, and logarithms with a base e are natural logarithms. On your calculator, the base 10 logarithm is noted by log, and the base e logarithm is noted by ln.
There are an infinite number of bases and only a few buttons on your calculator. You can convert a logarithm with a base that is not 10 or e to an equivalent logarithm with base 10 or e. If you are interested in a discussion on how to change the bases of a logarithm, click on Change of Base.
For a discussion of the relationship between the graphs of logarithmic functions and exponential functions, click on graphs.