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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?

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    $\begingroup$ 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. $\endgroup$ Commented May 26, 2020 at 22:49
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    $\begingroup$ 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. $\endgroup$
    – Blue
    Commented May 26, 2020 at 22:52
  • $\begingroup$ @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 $\endgroup$
    – imranfat
    Commented May 26, 2020 at 22:53
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    $\begingroup$ @imranfat A logarithm isn't an exponent, just like how a cube root isn't a cube. $\endgroup$ Commented May 26, 2020 at 23:28
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    $\begingroup$ @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.) $\endgroup$ Commented May 26, 2020 at 23:41

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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 $?^2 = 23$.

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