How do you solve 10^(1-4m)=18?

Apr 14, 2018

$\textcolor{b l u e}{m = \frac{1 - {\log}_{10} \left(18\right)}{4} \approx - 0.6381812628}$

Explanation:

Using logarithms to the base $10$.

From the laws of logarithms:

${\log}_{b} \left({a}^{c}\right) = c {\log}_{b} \left(a\right)$

And, if:

$y = {\log}_{b} a \iff {b}^{y} = a$

${\log}_{b} \left({b}^{y}\right) = {\log}_{b} \left(a\right)$

$y {\log}_{b} \left(b\right) = {\log}_{b} \left(a\right)$

y=(log_b(a))/(log_b(b)

But: $y = {\log}_{b} \left(a\right)$

log_b(a)=(log_b(a))/(log_b(b)

Rearranging:

${\log}_{b} \left(b\right) = \frac{{\log}_{b} \left(a\right)}{{\log}_{b} \left(a\right)} = 1$

Using these ideas:

${10}^{1 - 4 m} = 18$

Taking base 10 logarithms:

${\log}_{10} \left({10}^{1 - 4 m}\right) = {\log}_{10} \left(18\right)$

$\left(1 - 4 m\right) {\log}_{10} \left(10\right) = {\log}_{10} \left(18\right)$

$1 - 4 m = {\log}_{10} \left(18\right)$

$\textcolor{b l u e}{m = \frac{1 - {\log}_{10} \left(18\right)}{4} \approx - 0.6381812628}$