# How do you solve ln x + ln (x+5) = ln 14?

Dec 19, 2015

I found $x = 2$

#### Explanation:

Use the property of the log:
$\log x + \log y = \log \left(x \cdot y\right)$
to get:
$\ln \left[x \left(x + 5\right)\right] = \ln \left(14\right)$
for the logs to be equal the arguments must be equal as well, or:
$x \left(x + 5\right) = 14$
solve for $x$:
${x}^{2} + 5 x - 14 = 0$
${x}_{1 , 2} = \frac{- 5 \pm \sqrt{25 + 56}}{2} = \frac{- 5 \pm \sqrt{81}}{2} = \frac{- 5 \pm 9}{2} =$
${x}_{1} = - 7$ NO it gives you a negative argument in the original logs.
${x}_{2} = 2$ YES