# The Logarithm Rules

$\log_a(b)=c\hspace{10pt}\Longrightarrow\hspace{10pt}a^c=b$

$\log_a(x)+\log_a(y)=\log_a(xy)$

$\log_a(x)-\log_a(y)=\log_a(x/y)$

$\log_a(x^n)=n\log_a(x)$

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

$\log_a(x)=\frac{\log_b(x)}{\log_b(a)}$

Example 1Write the following as a single logarithm:

$3\log_5(2)+2\log_5(4)-\log_5(6)$

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

$\log_5(2^3)+\log_5(4^2)-\log_5(6)$

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

$\log_5\left(\frac{64}{3}\right)$

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

$\log_a\left(\frac{x^2\sqrt{y}}{z^5}\right)$

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

$\log_a\left(x^2\right)+\log_a\left(\sqrt{y}\right)-\log_a\left(z^5\right)$

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

$2\log_a(x)+\frac{1}{2}\log_a(y)-5\log_a\left(z\right)$

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