logo20137

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)



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.