# How do you evaluate log_(1/4)( 1/64)?

Sep 11, 2016

${\log}_{\frac{1}{4}} \left(\frac{1}{64}\right) = 3$

#### Explanation:

In log format like this one, the question being asked is

"To what power/index must $\frac{1}{4}$ be raised to give $\frac{1}{64}$?"

OR: "How can I make $\frac{1}{4}$ into $\frac{1}{64}$ using an index?"

${4}^{3} = 64 , \therefore {\left(\frac{1}{4}\right)}^{3} = \frac{1}{64}$

${\log}_{\frac{1}{4}} \left(\frac{1}{64}\right) = 3$

OR $\rightarrow$Log form and index form are interchangeable:

${\log}_{a} b = c \text{ " hArr" } {a}^{c} = b$

${\log}_{\frac{1}{4}} \left(\frac{1}{64}\right) = x \text{ " hArr " } {\left(\frac{1}{4}\right)}^{x} = \frac{1}{64}$

$x = 3$

It will be a distinct advantage in your work with logs and indices if you know all the powers of the numbers from 1 to 10 (up to 1,000) by heart.

Ie up to ${10}^{3} = 1 , 000$