# Question #60774

May 9, 2016

$x = \frac{5 + 5 {e}^{4}}{{e}^{4} - 1} \approx 5.187$

#### Explanation:

We will use the following properties of logarithms :

• $\ln \left(a\right) - \ln \left(b\right) = \log \left(\frac{a}{b}\right)$ for $a , b > 0$
• ${e}^{\ln} \left(a\right) = a$

with these, we have

$\ln \left(x + 5\right) - \ln \left(x - 5\right) = 4$

(note that as the $\ln$ function only takes positive values, we must restrict $x$ to be greater than $5$, or else we would have $x - 5 < 0$)

$\implies \ln \left(\frac{x + 5}{x - 5}\right) = 4$

$\implies {e}^{\ln \left(\frac{x + 5}{x - 5}\right)} = {e}^{4}$

$\implies \frac{x + 5}{x - 5} = {e}^{4}$

$\implies x + 5 = {e}^{4} x - 5 {e}^{4}$

$\implies {e}^{4} x - x = 5 + 5 {e}^{4}$

$\implies \left({e}^{4} - 1\right) x = 5 + 5 {e}^{4}$

$\therefore x = \frac{5 + 5 {e}^{4}}{{e}^{4} - 1} \approx 5.187$