Express as a single logarithm

Question

I am working on logarithms and have to express this as a single log: $2\log_a(x) - 3\log_a(y)$

I tried and came up with $$2\log_a\left(\frac x{\log_a(y)}\right)$$ but I'm not sure if I'm doing it right.

Answer 1

You've got the right idea, but we need to consider the coefficients, as well. That is, before applying the fact that $\log x - \log y = \log \frac xy$, we need to use the fact that $$c\log x = \log x^c$$

$$2 \log_a X - 3 \log_a Y = \log_a X^2 - \log_a Y^3 = \log_a\left(\frac{X^2}{Y^3}\right)$$