Question: If $2^x = 3^y = 12^z$, show that: $\dfrac{1}{z} = \dfrac{1}{y} + \dfrac{2}{x}$.

Solution:

Given: $2^x = 3^y = 12^z$

To prove: $\dfrac{1}{z} = \dfrac{1}{y} + \dfrac{2}{x}$

We know,

$2^x = 12^z$

$or, 2 = 12^{\dfrac{z}{x}}$ - (i)


Also,

$3^y = 12^z$

$or, 3 = 12^{\dfrac{z}{y}}$ - (ii)


Now,

$12^z = 12^z$

$or, 12^z = (3×4)^z$

$or, 12^z = (3×2²)^z$

$or, 12^z = 3^z × 2^{2z}$

[ Put value of 3 and 2 from equations (ii) and (i), respectively ]

$or, 12^z = \left ( 12^{\dfrac{z}{y}} \right )^z × \left ( 12^{\dfrac{z}{x}} \right )^{2z}$

$or, 12^z = 12^{\dfrac{z²}{y}} × 12^{\dfrac{2z²}{x}}$

$or, 12^z = 12^{\dfrac{z²}{y} + \dfrac{2z²}{x}}$

$or, z = \dfrac{z²}{y} + \dfrac{2z²}{x}$

$or, z = \dfrac{xz² + 2yz²}{xy}$

$or, z = \dfrac{z²(x+2y)}{xy}$

$or, \dfrac{xyz}{z²} = x +2y$

$or, \dfrac{xy}{z} = x + 2y$

[ Dividing by xy ]

$or, \dfrac{xy}{xyz} = \dfrac{x}{xy} + \dfrac{2y}{xy}$

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

#proved


Related Notes and Solutions:

Here is the website link to the notes of Indices.

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