2022-07-12

Something to the power of a logarithm
This is probably a very obvious question, but here goes...
An answer in my textbook claims that
${3}^{\mathrm{log}n}={n}^{\mathrm{log}3}$
and that
$4{n}^{2}\left(3/4{\right)}^{\mathrm{log}n}=4{n}^{\mathrm{log}3}$
Why, using more basic laws, is this the case?
(Unfortunately Google confuses this question with changing bases, exponentiation being the inverse of log (which is of course related), and similar matters.)

alomjabpdl0

Expert

Recall that $3={b}^{{\mathrm{log}}_{b}3}$
Therefore ${3}^{{\mathrm{log}}_{b}n}=\left({b}^{{\mathrm{log}}_{b}3}{\right)}^{{\mathrm{log}}_{b}n}={b}^{\left({\mathrm{log}}_{b}3\right)\left({\mathrm{log}}_{b}n\right)}=\left({b}^{{\mathrm{log}}_{b}n}{\right)}^{{\mathrm{log}}_{b}3}={n}^{{\mathrm{log}}_{b}3}$

cooloicons62

Expert

Hint : $a=b$ implies $\mathrm{log}a=\mathrm{log}b$ and $\mathrm{log}{a}^{b}=b\mathrm{log}a$