Logarithm

Submitted by alpha on Mon, 06/15/2009 - 06:06.

If $log_{12} 18=\alpha$ [log 18 to the base 12] and $log_{24} 54=\beta$ [log 54 to the base 24] , Prove that $\alpha\beta+5(\alpha-\beta)=1$

Please show your steps.

$log_2{3}=t$

$a=\frac{2t+1}{t+2}$

$b=\frac{3t+1}{t+3}$

$\frac{2t+1}{t+2}\frac{3t+1}{t+3}+5(\frac{2t+1}{t+2}-\frac{3t+1}{t+3})=1$