# How do you solve 5*11^(5a+10)=57?

Oct 16, 2016

$a = - 1.797$

#### Explanation:

$5 \cdot {11}^{5 a + 10} = 57$

$\frac{5 \cdot {11}^{5 a + 10}}{5} = \frac{57}{5} \textcolor{w h i t e}{a a a}$Divide both sides by 5

${11}^{5 a + 10} = \frac{57}{5}$

$\log \left({11}^{5 a + 10}\right) = \log \left(\frac{57}{5}\right) \textcolor{w h i t e}{a a a}$Take the log of both sides

$\left(5 a + 10\right) \log 11 = \log \left(\frac{57}{5}\right) \textcolor{w h i t e}{a a}$Use the log rule $\log {x}^{a} = a \log x$

$\frac{\left(5 a + 10\right) \log 11}{\log 11} = \frac{\log \left(\frac{57}{5}\right)}{\log 11} \textcolor{w h i t e}{a a}$Divide both sides by log11

$5 a + 10 = \frac{\log \left(\frac{57}{5}\right)}{\log 11}$

$\textcolor{w h i t e}{a a} - 10 \textcolor{w h i t e}{a a a} - 10 \textcolor{w h i t e}{a a a}$Subtract 10 from both sides

$5 a = \frac{\log \left(\frac{57}{5}\right)}{\log 11} - 10$

$\frac{5 a}{5} = \frac{\frac{\log \left(\frac{57}{5}\right)}{\log 11} - 10}{5} \textcolor{w h i t e}{a a a}$Divide both sides by 5

$a = - 1.797 \textcolor{w h i t e}{a a a}$Use a calculator