# Solve for x using properties of logarithms: ln(9)+ln(x+2)=2ln(x+2) ?

Feb 28, 2018

$x = 7$

#### Explanation:

$\ln \left(9\right) + 1 n \left(x + 2\right) = 2 \ln \left(x + 2\right)$

Subtract $2 \ln \left(x + 2\right)$ from both sides:

$\ln \left(9\right) + 1 n \left(x + 2\right) - 2 \ln \left(x + 2\right) = 0$

Simplify:

$\ln \left(9\right) - 1 n \left(x + 2\right) = 0$

$\ln \left(a\right) - \ln \left(b\right) = \ln \left(\frac{a}{b}\right)$

Hence:

$\ln \left(9\right) - 1 n \left(x + 2\right) = 0 \implies \ln \left(\frac{9}{x + 2}\right) = 0$

Raising the base $e$ to these powers:

e^(ln(9/(x+2))=e^0

$\frac{9}{x + 2} = 1$

$x = 7$

Substituting in original equation:

$\ln \left(9\right) + 1 n \left(\left(7\right) + 2\right) = 2 \ln \left(\left(7\right) + 2\right)$

$\ln \left(9\right) + 1 n \left(9\right) = 2 \ln \left(9\right)$

$2 \ln \left(9\right) = 2 \ln \left(9\right)$