Raul Walker

2022-07-14

How do you solve a logarithm with a non-integer base?
How would one calculate the log of a number where the base isn't an integer (in particular, an irrational number)? For example:

${\mathrm{log}}_{0.5}8=-3$
How would you solve this, and how would this work for an irrational base (like $\sqrt{2}$)?

vrtuljakc6

Expert

Let's rewrite this in a different way:
${0.5}^{x}=8$
Take the logarithm with respect to any base $a>0$
${\mathrm{log}}_{a}\left({0.5}^{x}\right)={\mathrm{log}}_{a}8$
which becomes
$x{\mathrm{log}}_{a}0.5={\mathrm{log}}_{a}8$
or
$x=\frac{{\mathrm{log}}_{a}8}{{\mathrm{log}}_{a}0.5}$
You would stop here weren't from the fact that $8={2}^{3}$ and $0.5={2}^{-1}$ so.
$x=\frac{{\mathrm{log}}_{a}8}{{\mathrm{log}}_{a}0.5}=\frac{{\mathrm{log}}_{a}{2}^{3}}{{\mathrm{log}}_{a}{2}^{-1}}=\frac{3{\mathrm{log}}_{a}2}{-{\mathrm{log}}_{a}2}=-3$
You need to compute no logarithm, actually.

invioor

Expert

This is the base changing formula :
${\mathrm{log}}_{a}\left(x\right)=\frac{{\mathrm{log}}_{b}\left(x\right)}{{\mathrm{log}}_{b}\left(a\right)}$

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