If $\log_{30}{3} = c$ and $\log_{30}{5} = d$ then the value of $\log_{30}{8} $ is??

Question

I attempted the following:

$\log_{30}{8} = 3\log_{30}{2}$

$\log_{30}{3} = c$ is equivalent to $3 = 30^c$

$\log_{30}{5} = d$ is equivalent to $5 = 30^d$

What should I do further?? Is there any basic formula for $log_a{(x - y)}$ ??

Answer 1

Here's an idea: $$ \log_{30} 8 = \log_{30}(2^3) = 3 \log_{30}2 = 3 \log_{30}\left(\frac{30}{15}\right) = 3 \log_{30}\left(\frac{30}{3 \cdot 5}\right) $$ I think you should be able to figure out the rest.