Evaluate each log without a calculator log_(4) (1)/(16) log_(2)\sqrt[5]{32}

Question

Evaluate each log without a calculator


`log_(4) (1)/(16)`


`log_(2)\sqrt[5]{32}`

Answer 1

QUESTION 1

The given logarithmic expression is

`\log_4((1)/(16))`

We rewrite
`(1)/(16)` in the index form to base 4.

This implies that;

`\log_4((1)/(16))=\log_4(4^(-2))`

We now apply the power rule:
`\log_a(m^n)=n\log_a(m^n)`.

`\log_4((1)/(16))=-2\log_4(4)`

Recall that logarithm of the base is 1.

`\log_4((1)/(16))=-2(1)`

`\log_4((1)/(16))=-2`

QUESTION 2

The given logarithm is;

`\log_2(\sqrt[5]{32})`

`\log_2(\sqrt[5]{2^5})`

This is the same as;

`\log_2(2^{5* (1)/(5)})`

`\log_2(2^(1))`

`\log_2(2)=1`