Question

How to express this equation in terms of m and n ?

Given `log_8 3 = m` and `log_8 5 = n`, express `log_3 50` in terms of m and n.

Answer 1

`log_3(50)=(1+6n)/(3m)`

Explanation:

We need to know the logarithm rules:

• `log_a(bc)=log_a(b)+log_a(c)`
• `log_a(b^c)=clog_a(b)`
• `log_a(b)=log_c(b)/log_c(a)`
• `log_a(a)=1`

Then:

`log_3(50)=log_8(50)/log_8(3)=log_8(50)/m`

Splitting up `50` into its prime factorization:

`=log_8(2*5^2)/m=(log_8(2)+log_8(5^2))/m=(log_8(8^(1/3))+log_8(5^2))/m`

`=(1/3log_8(8)+2log_8(5))/m=(1/3+2n)/m=(1+6n)/(3m)`