How do you solve log(5+x)-log(x-3)=log3?

Dec 18, 2015

$\textcolor{w h i t e}{\times} x = 7$

Explanation:

$\textcolor{w h i t e}{\times} \log \left(5 + x\right) - \log \left(x - 3\right) = \log 3$

The quotient of two positive numbers are calculated as difference of their logarithms. Therefore,
$\implies \log \left(\frac{5 + x}{x - 3}\right) = \log 3$
$\implies \frac{5 + x}{x - 3} = 3$
$\implies 3 x - 9 = 5 + x$

$\implies 3 x - 9 \textcolor{red}{- x + 9} = 5 + x \textcolor{red}{- x + 9}$
$\implies \textcolor{red}{\frac{1}{2} \times} 2 x = \textcolor{red}{\frac{1}{2} \times} 14$

$\implies x = 7$