# How do you solve Ln x - Ln(x+1) = 1?

Dec 1, 2016

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#### Explanation:

Use the subtraction rule of logarithms that ${\log}_{a} \left(n\right) - {\log}_{a} \left(m\right) = {\log}_{a} \left(\frac{n}{m}\right)$.

$\ln \left(\frac{x}{x + 1}\right) = 1$

$\frac{x}{x + 1} = {e}^{1}$

$x = e \left(x + 1\right)$

$x = e x + e$

$x - e x = e$

$x \left(1 - e\right) = e$

$x = \frac{e}{1 - e}$

If you want an approximation, $x \cong - 1.58$. However, since $x > 0$ for the natural logarithms to be defined, this solution is extraneous.

Hopefully this helps!