# How do you solve ln(3x+1)-ln(5+x)=ln2?

Oct 1, 2015

$x = 9$

#### Explanation:

$\ln \left(3 x + 1\right) - \ln \left(5 + x\right) = \ln \left(2\right) \implies$ using laws of logs:

$\ln \left[\frac{3 x + 1}{x + 5}\right] = \ln \left(2\right) \implies$ if $\ln \left(A\right) = \ln \left(B\right) \Leftrightarrow A = B$:

$\frac{3 x + 1}{x + 5} = 2 \implies$ multiply by $\left(x + 5\right)$:

$3 x + 1 = 2 \left(x + 5\right) \implies$ expand right side:

$3 x + 1 = 2 x + 10 \implies$ subtract$- 2 x$ and 1 from both sides:

$3 x - 2 x = 10 - 1 \implies$ simplify:

$x = 9$