# How do you solve 2lnx+3ln2=5?

Dec 15, 2015

$x \cong 4.30716$

#### Explanation:

Property of Logarithmic expression

$\log A + \log B = L o g \left(A B\right) \text{ " " " } \left(1\right)$
$n \log A = \log {A}^{n} \text{ " " } \left(2\right)$

Given :

$2 \ln x + 3 \ln 2 = 5$

Rewrite as:
Using rule (1)

$\ln {x}^{2} + \ln {2}^{3} = 5$

Using rule (1)

$\ln \left({x}^{2} \cdot 8\right) = 5$

Raise the expression to exponential form, with the base of $e$

e^(ln(8x^2) = e^5

$8 {x}^{2} = {e}^{5}$
${x}^{2} = \frac{{e}^{5}}{8}$

$x = \pm \sqrt{\frac{{e}^{5}}{8}}$

$x \cong 4.30716$

Because the argument of any logarithm always POSITIVE and greater than zero, due to domain restriction.