Evaluating Logarithmic Expressions

Question

Evaluate: $$\log_4 \left(\dfrac{1}{256}\right)$$

I am not sure how to approach this since there is nothing set equal to it.

Answer 1

Really it should say "evaluate" not "solve" since, as you note, there isn't an equation, per se. Note that $\frac{1}{256} = 4^{-4}$. Can you see how to proceed?

Answer 2

First note that $$ \log_b x^n=n\log_b x $$ And $$ \log_b b=1 $$ So now we have $$ \log_4\left(\frac{1}{256}\right) = \log_4\left(\frac{1}{4^4}\right)$$ $$= \log_4\left(4^{-4}\right)= -4\log_4\left(4\right)=-4 $$