1. ## Indices, logarithms question

hey there
been stuck on this one for ages. can't really get a grasp on it

Given: $3^x = 4^y = 12^z$ , show that $z = \frac{xy}{x+y}$

2. ## Re: Indices, logarithms question

Originally Posted by walleye
hey there
been stuck on this one for ages. can't really get a grasp on it

Given: $3^x = 4^y = 12^z$ , show that $z = \frac{xy}{x+y}$
Here's one way you can do it :
We know that $3^x = 4^y = 12^z$
now taking logarithms, this equation becomes $\implies x\log 3 = y\log 4 = z\log 12$
we also write: $x\log 3 = y\log 4 = z\log 12 = k$
now $\log 3=\frac{k}{x}$ and $\log 4=\frac{k}{y}$

Also, $z\log 12 = k \implies z\log(4.3)=k$
by applying the log properties: $\implies z\log 4 + z\log 3 = k \implies z(\log 4 +\log 3)=k$
putting in the values of $\log 4$ and $\log 3$, $\implies$ $zk(\frac{1}{x}+\frac{1}{y})=k$
$k$ gets cancelled,and simplifying gives us: $z(\frac{x+y}{xy})=1$
which gives $z=\frac{xy}{x+y}$