Hello, amandamandy!
I'll take a
guess at what you meant.
And if you don't know what a
cube root is, you need help . . .
Given: \(\displaystyle \log_r\left(\sqrt[3]{\frac{m^2r^5}{p}}\right)\)
We have: \(\displaystyle \,\log_r\left(\sqrt[3]{\frac{m^2r^5}{p}}\right)\)
\(\displaystyle \;\;\;=\;\log_r\left(\frac{m^2r^5}{p}\right)^{\frac{1}{3}}\)
\(\displaystyle \;\;\;=\;\frac{1}{3}\cdot\log_r\left(\frac{m^2r^5}{p}\right)\)
\(\displaystyle \;\;\;=\;\frac{1}{3}\left[\log_r(m^2)\,+\,\log_r(r^5)\,-\,\log_r(p)\right]\)
\(\displaystyle \;\;\;=\frac{1}{3}\left[2\cdot\log_r(m) \,+\,5\cdot\log_r(r)\,-\,\log_r(p)\right]\)
\(\displaystyle \;\;\;=\;\frac{1}{3}\left[2\cdot\log_r(m)\,+\,5\cdot1\,-\,\log_r(p)\right]\)
\(\displaystyle \;\;\;=\;\frac{1}{3}\left[2\cdot\log_r(m)\,+\,5\,-\,\log_p(p)\right]\)