emmostatwf

2022-09-09

Evaluating Logarithmic Expressions

Evaluate:

$${\mathrm{log}}_{4}\left({\displaystyle \frac{1}{256}}\right)$$

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

Evaluate:

$${\mathrm{log}}_{4}\left({\displaystyle \frac{1}{256}}\right)$$

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

Leslie Braun

Beginner2022-09-10Added 7 answers

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?

st3he1d0t

Beginner2022-09-11Added 3 answers

First note that

$${\mathrm{log}}_{b}b=1$$

So now we have

$${\mathrm{log}}_{4}\left(\frac{1}{256}\right)={\mathrm{log}}_{4}\left(\frac{1}{{4}^{4}}\right)$$

$$={\mathrm{log}}_{4}\left({4}^{-4}\right)=-4{\mathrm{log}}_{4}\left(4\right)=-4$$

$${\mathrm{log}}_{b}b=1$$

So now we have

$${\mathrm{log}}_{4}\left(\frac{1}{256}\right)={\mathrm{log}}_{4}\left(\frac{1}{{4}^{4}}\right)$$

$$={\mathrm{log}}_{4}\left({4}^{-4}\right)=-4{\mathrm{log}}_{4}\left(4\right)=-4$$

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