Question

Answer for this for class 9

Answer 1

${2}^{x} = {3}^{y} = {6}^{z}$
Take log on all three sides.
$x log(2) = y log(3) = z log(6)$
Compare the terms with X and Z$x log(2) = z log(6) \\ log(2) = \frac{z}{x} log(6)$
Compare the terms with Y and Z
$y log(3) = z log(6) \\ log(3) = \frac{z}{y} log(6)$
Add these two equations.
$log(2) + log(3) = ( \frac{z}{x} + \frac{z}{y} ) log(6) \\ log(6) = ( \frac{z}{x} + \frac{z}{y} ) log(6)$
Now we can cancel out {log(6)} from both sides.
$\frac{z}{x} + \frac{z}{y} = 1 \\ z( \frac{x + y}{xy}) = 1 \\ z = \frac{xy}{x + y}$
Hence Proved