log base r index:3 square root of m^2 * r^5 /p

[Guest]

Given: log base r index:3 square root of m^2 * r^5 /p

I got as far as 1/2log base r of m + 5/3 log base r of r - log base r of p

does anyone know how to simplify it further?

 

[stapel]

Re: properties of log

Given: log base r index:3 square root of m^2 * r^5 /p

I'm sorry, but I can't make heads or tails of this. What do you mean by "index:3"? Is everything after the "log base r" included inside the log's argument? How much is inside the square root?

It might help if you used standard notation. The link to "Karl's Notes" (in the "Forum Help" pull-down menu at the very top of this page) explains notation that everybody else will understand. But at the least, please use grouping symbols.

When you reply, please also include the instructions. Thank you.

Eliz.

 

[soroban]

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: \(\displaystyle \,\log_r\left(\sqrt[3]{\frac{m^2r^5}{p}}\right)\)

\(\displaystyle \;\;\;=\;\log_r\left(\frac{m^2r^5}{p}\right)^{\frac{1}{3}}\)

\(\displaystyle \;\;\;=\;\frac{1}{3}\cdot\log_r\left(\frac{m^2r^5}{p}\right)\)

\(\displaystyle \;\;\;=\;\frac{1}{3}\left[\log_r(m^2)\,+\,\log_r(r^5)\,-\,\log_r(p)\right]\)

\(\displaystyle \;\;\;=\frac{1}{3}\left[2\cdot\log_r(m) \,+\,5\cdot\log_r(r)\,-\,\log_r(p)\right]\)

\(\displaystyle \;\;\;=\;\frac{1}{3}\left[2\cdot\log_r(m)\,+\,5\cdot1\,-\,\log_r(p)\right]\)

\(\displaystyle \;\;\;=\;\frac{1}{3}\left[2\cdot\log_r(m)\,+\,5\,-\,\log_p(p)\right]\)