# How do you solve for x? Give any approximate result to 3 significant digits. Check your answers.

## $3 \log x - 1 = 3 \log \left(x - 1\right)$

Nov 28, 2016

$x = 1.866$

#### Explanation:

Start by doing a little bit of algebra to isolate the logs on one side of the equation.

$3 \log x - 3 \log \left(x - 1\right) = 1$

$3 \left(\log x - \log \left(x - 1\right)\right) = 1$

$\log x - \log \left(x - 1\right) = \frac{1}{3}$

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

$\log \left(\frac{x}{x - 1}\right) = \frac{1}{3}$

$\frac{x}{x - 1} = {10}^{\frac{1}{3}}$

$\frac{x}{x - 1} = \sqrt[3]{10}$

$x = \sqrt[3]{10} \left(x - 1\right)$

$x = \sqrt[3]{10} x - \sqrt[3]{10}$

$x - \sqrt[3]{10} x = - \sqrt[3]{10}$

$x \left(1 - \sqrt[3]{10}\right) = - \sqrt[3]{10}$

$x = - \frac{\sqrt[3]{10}}{1 - \sqrt[3]{10}}$

$x = 1.866$

Hopefully this helps!