Question

if 3^x=5^y=75^z then prove (1/x+2/y=1/z)​

Answer 1

EXPLANATION.

$\implies 3^{x} \ = 5^{y} \ = 75^{z}$

As we know that,

Let their term is equal to any constant value = k.

$\implies 3^{x} \ = 5^{y} \ = 75^{z} \ = k.$

$\implies 3^{x} \ = k$

$\implies 3 = k^{(1/x)} . - - - - - (1).$

$\implies 5^{y} \ = k$

$\implies 5 = k^{(1/y)} . - - - - - (2).$

$\implies 75^{z} \ = k$

$\implies 75 \ = k^{(1/z)} . - - - - - (3).$

As we know that.

⇒ 75 = 5² x 3.

We can write equation as,

$\implies 5^{2} \times 3 = k^{(1/z)} .$

Put the values in the equation, we get.

$\implies \bigg(k^{(1/y)} \bigg)^{2} \times \bigg(k^{(1/x)} \bigg) \ = \bigg(k^{(1/z)} \bigg)$

$\implies k^{(2/y)} \times k^{(1/x)} \ = k^{(1/z)}$

$\implies k^{(2/y) + (1/x)} \ = k^{(1/z)}$

$\implies \dfrac{2}{y} \ + \dfrac{1}{x} \ = \dfrac{1}{z}$

Hence Proved.