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=(log_xb)/(log_xa)`

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

`log_(27)729=(log_3729)/(log_327)`
`\begin{gathered} 729=3^6 \\ 27=3^3 \end{gathered}`

Therefore,

`\begin{gathered} log_(27)729=(log_3729)/(log_327) \\ log_(27)729=(log_33^6)/(log_33^3) \end{gathered}`

Given the log rule below

`logb^a=alogb`

The given logarithms becomes

`log_(27)729=(6log_33)/(3log_33)`

Given the log rule below

`log_aa=1`

Apply the rule to the given log

`\begin{gathered} log_(27)729=(6*1)/(3*1) \\ log_(27)729=(6)/(3) \\ log_(27)729=2 \end{gathered}`

Hence, the solution to the given logarithm is 2