Question

Simplify the following expression using the change of base formula:

Answer 1

Given: The logarithm below

$log_{27}729$

To Determine: The solution of the given logarithm

Solution

Using change of base formula

$log_ab=\frac{log_xb}{log_xa}$

Let the common base be 3. Therefore, the given logarithm becomes

$log_{27}729=\frac{log_3729}{log_327}$$\begin{gathered} 729=3^6 \\ 27=3^3 \end{gathered}$

Therefore,

$\begin{gathered} log_{27}729=\frac{log_3729}{log_327} \\ log_{27}729=\frac{log_33^6}{log_33^3} \end{gathered}$

Given the log rule below

$logb^a=alogb$

The given logarithms becomes

$log_{27}729=\frac{6log_33}{3log_33}$

Given the log rule below

$log_aa=1$

Apply the rule to the given log

$\begin{gathered} log_{27}729=\frac{6\times1}{3\times1} \\ log_{27}729=\frac{6}{3} \\ log_{27}729=2 \end{gathered}$

Hence, the solution to the given logarithm is 2