Question

Evaluate without a calculator.
`log_(4)(16) + log_(3)(27)`please show your work..!

Answer 1

The value of the given expression is;

`5`

Step-by-step explanation:

Given the logarithmic expression;

`\log _416+\log _327`

To solve, Recall the rules of logarithm;

`\begin{gathered} \log _ax^y=y*\log _ax\text{ ------1} \\ \log _aa=1\text{ -------2} \end{gathered}`

Apply the above rules;

`\begin{gathered} \log _416+\log _327 \\ \log _44^2+\log _33^3 \\ \text{Applying rule 1;} \\ 2\log _44+3\log _33 \end{gathered}`

Aplplying rule 2;

`\begin{gathered} 2\log _44+3\log _33 \\ =2(1)+3(1) \\ =2+3 \\ =5 \\ \log _416+\log _327=5 \end{gathered}`
`\begin{gathered} \text{Note that from the rule 2;} \\ \log _aa=1 \\ so\text{ when a logarithm has the same number as the base it will equal to 1;} \\ \text{for example;} \\ \log _44=1 \\ \log _33=1 \\ \log _55=1 \end{gathered}`

Therefore, the value of the given expression is;

`5`