Question

Evaluate the following expression using the properties of logarithms.

log subscript 4 left parenthesis 4 right parenthesis plus log subscript 4 left parenthesis 16 right parenthesis plus log subscript 4 left parenthesis 64 right parenthesis


10


9


6


17

Answer 1

6

Explanation:

Given logarithmic expression:

`\log_(4)\left(4\right)+\log_(4)\left(16\right)+\log_(4)\left(64\right)`

Rewrite 16 as 4² and 64 as 4³:

`\implies \log_(4)\left(4\right)+\log_(4)\left(4\right)^2+\log_(4)\left(4\right)^3`

`\textsf{Apply the log power law}: \quad \log_ax^n=n\log_ax`

`\implies \log_(4)\left(4\right)+2\log_(4)\left(4\right)+3\log_(4)\left(4\right)`

`\textsf{Apply log law}: \quad \log_aa=1`

`\implies 1+2(1)+3(1)`

Simplify:

`\implies 1+2+3`

`\implies 3 + 3`

`\implies 6`