Question

Given 2^x=3^y=6^z find z in terms of x and y

Answer 1

Given -

${2}^{x} = {3}^{y} = {6}^{z}$

To find -

The value of z

Solution -

$Let \: \: {2}^{x} = {3}^{y} = {6}^{z} = k$

Then,

$\implies{2}^{x} = k$

$⟹\: \: 2 = {k}^{ \frac{1}{x} } ........1)$

And,

$⟹{3}^{y} = k$

$⟹3 = {k}^{ \frac{1}{y} } ......2)$

And,

$⟹{6}^{z} = k$

$⟹6 = {k}^{ \frac{1}{z} }$

$⟹2 \times 3 = {k}^{ \frac{1}{z} }$

$⟹ {k}^{ \frac{1}{x} } \times {k}^{ \frac{1}{y} } = {k}^{ \frac{1}{z} } (by \: 1 \: and \: 2)$

$⟹{k}^{ \frac{1 }{x} + \frac{1}{y} } = {k}^{ \frac{1}{z} }$

$⟹\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$

$⟹\frac{y + x}{xy} = \frac{1}{z}$

By cross multiplying -

$⟹z(y + x) = xy$

$⟹z = \frac{xy}{y + x}$

Therefore the value of z is xy/y+x