Question

if knowing then tell​

Answer 1

Given -

${3}^{x} = {5}^{y} = {(75)}^{z}$

To show -

$z = \frac{xy}{2x + y}$

Solution -

$Let, </p><p>\: {3}^{x} = {5}^{y} = {(75)}^{z} = k$

${3}^{x} = k$

$3 = {k}^{ \frac{1}{x} } ........1)$

and,

${5}^{y} = k$

$5 = {k}^{ \frac{1}{y} } ......2)$

and,

=>${(75)}^{z} = k$

=>$75 = k {}^{ \frac{1}{z} }$

=>$25 \times 3 = {k}^{ \frac{1}{z} }$

=>${(5)}^{2} \times 3 = {k}^{ \frac{1}{z} }$

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

=>${k}^{ \frac{2}{y} } \times {k}^{ \frac{1}{x} } = {k}^{ \frac{1}{z} }$

=>${k}^{ \frac{2}{y} + \frac{1}{x} } = {k}^{ \frac{1}{z} }$

=>${k}^{ \frac{2x + y}{xy} } = {k}^{ \frac{1}{z} }$

=>$\frac{2x + y}{xy} = \frac{1}{z}$

By cross multiplying,

=>$z(2x + y) = xy$

=>$z = \frac{xy}{2x + y}$

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