# How do you solve lnx-ln3=2?

Feb 18, 2016

$x = 3 \cdot {e}^{2}$

#### Explanation:

from the given: $\ln x - \ln 3 = 2$

$\ln \left(\frac{x}{3}\right) = 2$

also this means

${\ln}_{e} \left(\frac{x}{3}\right) = 2$

taking the exponential form

${e}^{2} = \frac{x}{3}$

$3 \cdot {e}^{2} = 3 \cdot \frac{x}{3}$ multiplying both sides by 3

$3 \cdot {e}^{2} = \cancel{3} \cdot \frac{x}{\cancel{3}}$

$3 \cdot {e}^{2} = x$

and

$\textcolor{red}{x = 3 \cdot {e}^{2}}$

Check: at $x = 3 \cdot {e}^{2}$ using the original equation

$\ln x - \ln 3 = 2$

$\ln 3 \cdot {e}^{2} - \ln 3 = 2$

$\ln \left(\frac{3 \cdot {e}^{2}}{3}\right) = 2$

$\ln \left({e}^{2}\right) = 2$

$2 = 2$