Isn't square root a bit like Log()?

I want start by saying that my math skills aren't great and I'm trying to learn.

I took a look at square root.

Squaring the number means x^2.

And if I understood the square root correctly it does a bit inverse of squaring a number and gets back the x.

I had a friend tell me a while ago that Log() is also opposite of exponent, wouldn't that mean that square root is like a variant of Log() that only inverse a squared number?

• the inverse of $x^2$ is a the square root, the inverse of $2^x$ is the (base 2) logarithm. The inverse of $x^a$ is the $a$-th root, the inverse of $a^x$ is the (base a) logarithm. May 26, 2020 at 22:49
• In a certain light, yes: roots and logs are kinda the same. They both look at the relation $a^b=c$ and ask "If I know one of the numbers on the left (and the number on the right), then what's the other number?" ... When you know $b$ (and $c$) and seek $a$, you get what we call "the $b$-th root of $c$"; when you know $a$ (and $c$) and seek $b$, you get what we call "the base-$a$ logarithm of $c$". ... Interestingly, a popular question (from 2011!) on Math.SE asked for ideas about how to make this interplay clearer in the notation we use.
– Blue
May 26, 2020 at 22:52
• @JensRenders Of course it is. Here is a link, but you being a master's that link ought not to be needed: columbia.edu/itc/sipa/math/logarithms.html May 26, 2020 at 22:53
• @imranfat A logarithm isn't an exponent, just like how a cube root isn't a cube. May 26, 2020 at 23:28
• @imranfat (When you phrase it the way you did, it strongly suggests exponential function, which it's of course not. I think you know this, though.) May 26, 2020 at 23:41

It might be best to answer with some examples.

• The number $$\log_7 23$$ answers the question $$7^{?} = 23$$.

• The number $$\sqrt[7]{23}$$ answers the question $$?^7 = 23$$.

As you can see, these are different questions.

Some further points:

• When we write $$\log$$, this is short for $$\log_{10}$$. So the number $$\log 23$$ answers the question $$10^? = 23$$.

• When we write $$\sqrt{\phantom{x}}$$, this is short for $$\sqrt[2]{\phantom{x}}$$. So the number $$\sqrt{23}$$ answers the question $$?^2 = 23$$.