Why are logarithms and exponentials inverse? 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.
 A: They are defined that way; the log function is defined to be the inverse of the exponential.  
A: 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.
