# How do you solve  logx + log(x + 15) = 2?

Mar 21, 2016

#### Explanation:

$\log x + \log \left(x + 15\right) = 2$
$\log \left(x \left(x + 15\right)\right) = 2$

Here is the catch. This log can be from any base p. I assume this is log base 10. So
${\log}_{10} \left({x}^{2} + 15 x\right) = 2$
Taking antilog on both sides
${x}^{2} + 15 x = {10}^{2}$
${x}^{2} + 15 x - 100 = 0$

Then factorize this to get the solution.

$\left(x + 20\right) \setminus \times \left(x - 5\right) = 0$
$x = - 20 , 5$

Similar equations can be derived for various bases to get different solutions.

For general case the equation to solve is
${x}^{2} + 15 x - {p}^{2} = 0$
$x = \frac{- 15 \setminus \pm \sqrt{225 + 4 {p}^{2}}}{2}$