# How do you solve log (x – 2) + log x = log 3?

Nov 28, 2015

This equation has one solution $x = 3$

#### Explanation:

We start with: $\log \left(x - 2\right) + \log x = \log 3$

Before calculating $x$ we have to find the domain of the equation.
Since $\log$ is only defined for positive values we have to find out where $x - 2 > 0$, so we get the domain: D=(2;+oo)

$\log x \cdot \left(x - 2\right) = \log 3$

$x \cdot \left(x - 2\right) = 3$

${x}^{2} - 2 x = 3$

${x}^{2} - 2 x - 3 = 0$

$\Delta = 4 - 4 \cdot 1 \cdot \left(- 3\right) = 4 + 12 = 16$

$\sqrt{\Delta} = 4$

${x}_{1} = \frac{2 - 4}{2} = - 1$

${x}_{2} = \frac{2 + 4}{2} = 3$

From those values only ${x}_{2}$ is in the domain $D$ so it is the only solution of the equation.

Answer: $x = 2$