iristh3virusoo2

2022-02-23

I am trying to convert part of an equation from its log form into a linear form. Specifically, I am trying to convert ${10}^{4\mathrm{log}\left(x\right)}$, into ${x}^{4}$, but I'm really unsure of how to get from this first stage to the second. My experience with logarithms and exponents is limited, though I believe that ${10}^{4\mathrm{log}\left(x\right)}$ can be re-written as ${10}^{4}+{10}^{\mathrm{log}\left(x\right)}$, but I'm not sure that this helps my plight! Any very basic guidance would be greatly appreciated.

### Answer & Explanation

mastifo5h

A cool rule about logarithmic functions is that
$a\cdot {\mathrm{log}}_{b}\left(x\right)={\mathrm{log}}_{b}\left({x}^{a}\right)$.
Fron this,
${10}^{4\mathrm{log}\left(x\right)}={10}^{\mathrm{log}\left({x}^{4}\right)}$.
If we assume that you have a logarithm to the base 10, then
${10}^{{\mathrm{log}}_{10}\left({x}^{4}\right)}={x}^{4}$.

Taking logarithms,
$\mathrm{log}\left({10}^{4\mathrm{log}x}\right)=4\mathrm{log}x\mathrm{log}10=\mathrm{log}x\left(4\mathrm{log}10\right)=\mathrm{log}\left({x}^{4\mathrm{log}10}\right),$,
so you can exponentiate both sides to get to
${10}^{4\mathrm{log}\left\{x\right\}}={x}^{4\mathrm{log}\left\{10\right\}}$.
I shall spare you the rant about using log to mean ${\mathrm{log}}_{10}$.

Do you have a similar question?

Recalculate according to your conditions!