The Logarithm Rules





You can change the base of the log using the following formula:


Example 1Write the following as a single logarithm:



Using the rule \log_a(x^n)=n\log_a(x), the expression can be written as


Then, using the rule \log_a(x)+\log_a(y)=\log_a(xy), this expression can be written as

\log_5(2^3\times 4^2)-\log_5(6)

Finally, using \log_a(x)-\log_a(y)=\log_a(x/y), we can write it as

\log_5\left(\frac{2^3\times 4^2}{6}\right)

This simplifies to


Example 2Expand the following in terms of logs of x, y and z:


Using the rules for adding and subtracting logs with the same base, we can expand the expression as follows:


Using the rule for dealing with powers inside the log, this becomes:


Note that \sqrt{y}=y^{\frac{1}{2}}, hence the half in the final expression.