How do you condense 5ln(x - 2) - ln(x + 2) + 3lnx?

May 14, 2016

Remember that logs are just indices, therefore all the laws of indices apply..

Say the following out loud: If you are multiplying and dividing numbers, then you will add or subtract the logs.

Here the opposite is true.. If you are adding and subtracting logs, you multiply or divide the numbers.

$5 \ln \left(x - 2\right) - \ln \left(x + 2\right) + 3 \ln x$

= $\ln {\left(x - 2\right)}^{5} - \ln \left(x + 2\right) + {x}^{3}$

=$\ln \left[\frac{{\left(x - 2\right)}^{5} \left({x}^{3}\right)}{x + 2}\right] \text{ 3 log terms have become log of 1 term}$

=$\ln \left[\frac{{x}^{3} {\left(x - 2\right)}^{5}}{x + 2}\right]$