# How do you solve log x - log(x-10)=1?

Aug 10, 2015

$\textcolor{red}{x = \frac{100}{9}}$

#### Explanation:

 logx-log(x−10)=1

Recall that $\log a - \log b = \log \left(\frac{a}{b}\right)$, so

 logx-log(x−10)=log(x/(x-10))

$\log \left(\frac{x}{x - 10}\right) = 1$

Convert the logarithmic equation to an exponential equation.

${10}^{\log \left(\frac{x}{x - 10}\right)} = {10}^{1}$

Remember that ${10}^{\log} x = x$, so

$\frac{x}{x - 10} = 10$

$x = 10 \left(x - 10\right)$

$x = 10 x - 100$

$9 x = 100$

$x = \frac{100}{9}$

Check:

 logx-log(x−10)=1

If $x = \frac{100}{9}$,

 log(100/9)-log(100/9−10)=1

$\log \left(\frac{100}{9}\right) - \log \left(\frac{100}{9} - \frac{90}{9}\right) = 1$

$\log \left(\frac{100}{9}\right) - \log \left(\frac{100 - 90}{9}\right) = 1$

$\log \left(\frac{100}{9}\right) - \log \left(\frac{10}{9}\right) = 1$

$\log \left(\frac{\frac{100}{\textcolor{red}{\cancel{\textcolor{b l a c k}{9}}}}}{\frac{10}{\textcolor{red}{\cancel{\textcolor{b l a c k}{9}}}}}\right) = 1$

$\log \left(\frac{100}{10}\right) = 1$

$\log 10 = 1$

$1 = 1$

$x = \frac{100}{9}$ is a solution.