Why are logarithms and exponentials inverse?

Question

I'm unclear on why ${{{\log_bn=x}}}$ is the inverse of $b^x=n$, other than the fact someone told me that they were.

Answer 1

Logs were created so that multiplication could be done with addition. This means that $\log(ab) =\log(a)+\log(b) $.

Therefore, from a table of logs, look up the logs of $a$ and $b$, add them, and then do an inverse lookup to find the value of $ab$.

To make the inverse lookup easier, tables of the inverse function were created, so that, given $\log(c)$, $c$ could be found. If we call this function $alog$, for $arclog$ or inverse $\log$, then $alog(log(x)) = x$ and $\log(alog(x)) = x$.

This means that $alog(\log(a)+\log(b)) =ab $.

It turns out that, if $v$ is the value such that $\log(v) = 1$, then $\log(v^x) =x \log(v) = x $. This means that taking logs and raising $v$ to a power were inverse operations. Explicitly, $v^x = alog(x)$.

Similarly, it is easy to verify that $\log(a^b) =b\log(a), $ so $\log(\log(a^b)) =\log(b\log(a)) =\log(b)+\log(\log(a)) $. Therefore, $a^b =alog(alog(\log(b)+\log(\log(a)))) $.

And they all computed happily ever after.

Answer 2

They are defined that way; the log function is defined to be the inverse of the exponential.