# How do you solve  ln (x – 2) + ln (x + 2) = ln 5?

Jun 9, 2016

x=-3 or x=3

#### Explanation:

Using the property that says :

$\ln \left(a\right) + \ln \left(b\right) = \ln \left(a \cdot b\right)$
We have:

$\ln \left(x - 2\right) + \ln \left(x + 2\right) = \ln 5$

$\ln \left(\left(x - 2\right) \cdot \left(x + 2\right)\right) = \ln 5$

Rasing exponential both sides we will have:

$\left(x - 2\right) \cdot \left(x + 2\right) = 5$

Applying polynomial property on the equation above that says:

${a}^{2} - {b}^{2} = \left(a - b\right) \cdot \left(a + b\right)$

We have: $\left(x - 2\right) \cdot \left(x + 2\right) = {x}^{2} - 4$

So,
${x}^{2} - 4 = 5$
${x}^{2} - 4 - 5 = 0$
${x}^{2} - 9 = 0$
$\left(x - 3\right) \cdot \left(x + 3\right) = 0$

So,
$x - 3 = 0$ thus $x = 3$

Or,

$x + 3 = 0$ thus $x = - 3$