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

Jun 22, 2016

Use a few properties of logs to condense $\ln x + \ln \left(x - 2\right) - 5 \ln y$ into $\ln \left(\frac{{x}^{2} - 2 x}{{y}^{5}}\right)$.

Explanation:

Begin by using the property $\ln a + \ln b = \ln a b$ on the first two logs:
$\ln x + \ln \left(x - 2\right) = \ln \left(x \left(x - 2\right)\right) = \ln \left({x}^{2} - 2 x\right)$

Now use the property $a \ln b = \ln {b}^{a}$ on the last log:
$5 \ln y = \ln {y}^{5}$

Now we have:
$\ln \left({x}^{2} - 2 x\right) - \ln {y}^{5}$

Finish by combining these two using the property $\ln a - \ln b = \ln \left(\frac{a}{b}\right)$:
$\ln \left({x}^{2} - 2 x\right) - \ln {y}^{5} = \ln \left(\frac{{x}^{2} - 2 x}{{y}^{5}}\right)$