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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