Question

Evaluate each log without a calculator

$log_{4} \frac{1}{16}$

$log_{2}\sqrt[5]{32}$

Answer 1

QUESTION 1

The given logarithmic expression is

$\log_4(\frac{1}{16})$

We rewrite $\frac{1}{16}$ in the index form to base 4.

This implies that;

$\log_4(\frac{1}{16})=\log_4(4^{-2})$

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

$\log_4(\frac{1}{16})=-2\log_4(4)$

Recall that logarithm of the base is 1.

$\log_4(\frac{1}{16})=-2(1)$

$\log_4(\frac{1}{16})=-2$

QUESTION 2

The given logarithm is;

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

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

This is the same as;

$\log_2(2^{5\times \frac{1}{5}})$

$\log_2(2^{1})$

$\log_2(2)=1$