Find the value of log base two of the fourth root of 256, giving your answer in its simplest form.
The first thing that we can do is to figure out what the fourth root of 256 is. 256 can be written as four to the fourth power. If we would take the fourth root of something to the fourth power, it would disappear, so we can replace the fourth root of 256 with four.
There’re a few ways to solve this from here. Now since we have a base two, it might be useful to make four have a base of two, so instead of four, we can replace it with two squared. And when you have an exponent, you can bring it to the front.
Now from here, we can simplify. Log base two of two is just equal to one using this property. And the reason why if we would use a circle method, which isn’t an actual name for a certain method — it’s just a way to look at it — 𝑎 to the first power equals 𝑎, and that’s true.
So again, we can replace log base two of two with one, and two times one would give us two. An alternative route is we can set this equal to 𝑥, and using that same circle method, two to the 𝑥 power equals two squared.
Essentially, we’re just going from logarithmic form to exponential form when I say the circle method. So two to the 𝑥 power is equal to two to the second power, which would mean 𝑥 will equal two, just like we got.