Logarithms  Change of Base Formula
Most calculators have a
key and a
key. These keys allow us
to calculate logs with base 10 and base e, respectively.
To find the log when the base is a number other than 10 or e, we will use
the change of base formula.
Formula â€” Change of Base Formula
Here, b, c and x are positive real numbers. Also, b
≠ 1 and c ≠ 1.
Note:Notice that x is higher than b on both sides
of the equals sign in the change of base
formula.
In the change of base formula, we typically let the new base, c, be either
10 or e. This allows us to use a calculator to do the computations.
In the next example, we will use the change of base formula and show the
solution using both common logs (base 10) and natural logs (base e).
Example
Find: log_{7} 34
SolutionThe base of log7 34 is not 10 or e so we will use the change of base
formula. We will use common logs.
Change of base formula.


Replace b with 7, x with 34, and c with 10.


Use a calculator to find the two common logs. 

Simplify. 
≈ 1.812190828 
So, log_{7} 34 ≈ 1.81 (rounded to two decimal places).
Note:
We get the same result if we use natural
logs instead of common logs.
