# How do you solve 13^(x-3) = 7^(-8x)?

Mar 5, 2016

x = $\frac{3 \log 13}{\log 13 + 8 \log 7}$ = 0.424374, nearly.

#### Explanation:

Equate logarithms ( of any base ) and solve the resulting linear equation in x.

Mar 5, 2016

$\textcolor{red}{\text{ A very detailed explanation!}}$

$\text{ "x=(3log(13))/(8log(91))" "->" } \textcolor{b l u e}{x = \frac{3 \textcolor{w h i t e}{.} \sqrt[91]{13}}{8}}$

$\textcolor{g r e e n}{\approx 0.2132 \text{ to 4 decimal places}}$

#### Explanation:

$\textcolor{b l u e}{\text{Principle used to solve this problem}}$

If you have $\log \left({a}^{b}\right)$ then this can be written as $b \log \left(a\right)$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Given:$\text{ } {13}^{x - 3} = {7}^{- 8 x}$

$\textcolor{b r o w n}{\text{Take logs of both sides:}}$

$\text{ "log(13^(x-3))" " =" } \log \left({7}^{- 8 x}\right)$

$\text{ "=> (x-3)log(13)" "=" } - 8 x \log \left(7\right)$

$\textcolor{b r o w n}{\text{Divide both sides by } \log \left(13\right)}$

" "x-3" "=" "-8x xx(log(7))/(log(13)

$\textcolor{b r o w n}{\text{Divide both sides by } - 8 x}$

" "x/(-8x) -3/(-8x)" "=" "(log(7))/(log(13)

" "-1/8 +3/(8x)" "=" "(log(7))/(log(13)

$\textcolor{b r o w n}{\text{Add "1/8" to both sides}}$

$\text{ "3/(8x)" "=" } \frac{\log \left(7\right)}{\log \left(13\right)} + \frac{1}{8}$

$\textcolor{b r o w n}{\text{Divide both sides by 3}}$

$\text{ "1/(8x)" " =" } \frac{\log \left(7\right)}{3 \log \left(13\right)} + \frac{1}{24}$

$\textcolor{b r o w n}{\text{Multiply both sides by 8}}$

$\text{ "1/x" "=" } \frac{8 \log \left(7\right)}{3 \log \left(13\right)} + \frac{8}{24}$

$\textcolor{b r o w n}{\text{But } \frac{8}{24} = \frac{1}{3}}$

$\text{ } \frac{1}{x} = \frac{8 \log \left(7\right)}{3 \log \left(13\right)} + \frac{1}{3}$

$\text{ " 1/x=(8log(7)+log(13))/(3log(13))" "->"Corrected at this point}$

$\textcolor{b r o w n}{\text{Inverting everything}}$

" "x= (3log(13))/(8log(7)+log(13)

$\textcolor{w h i t e}{.}$

$\textcolor{R e d}{\text{Corrected value for } x}$
$x \approx \frac{3.3418}{7.8747} = 0.4244$ to 4 decimal places