Help with Logarithms. MEDAL and FAN. Question below...

Question

4everaddicted2anime · Answer

$$WhatPower _{2}(\frac{ 1 }{ 4 })=?$$

4everaddicted2anime · Answer

Is it 8?

4everaddicted2anime · Answer

@zepdrix

zepdrix · Answer

You have to turn the 1/4 into a 2 somehow.
It will require the negative exponent rule since we're dividing by 4 which is several 2's.

zepdrix · Answer

$$\large\rm \frac14=\frac{1}{2^2}=?$$

4everaddicted2anime · Answer

$$\frac{ 1 }{ 4 }$$

4everaddicted2anime · Answer

oh. So if the fraction is in the parenthesis then I need to put the log base with a numerator of 1?

4everaddicted2anime · Answer

Then I need to find to power which will get the denominators to be the same?

zepdrix · Answer

No, this isn't going to work out like the last problem.
You are correct that we can rewrite the square on the outside of some brackets,$$\large\rm \frac1{2^2}=\frac{1^2}{2^2}=\left(\frac{1}{2}\right)^2$$But we don't want to do that.
We're not trying to create a 1/2.
We're trying to create a 2.

zepdrix · Answer

I'm not sure if that's exactly what you were saying :)
But hopefully it explains some things.

zepdrix · Answer

We want to apply our exponent rule,$$\large\rm \frac{1}{2\cdot2}=\frac{1}{2^2}=2^{-2}$$We're dividing by two 2's.
So we can apply our exponent rule and put a -2 to show that we're dividing by 2 of them.

4everaddicted2anime · Answer

So the -2 is the answer?

zepdrix · Answer

$$\large\rm \log_2\left(\frac14\right)\quad=\log_2\left(2^{-2}\right)\quad=-2\cdot \log_2(2)\quad=-2$$Yes.

4everaddicted2anime · Answer

tysm