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Question : 85 of 86

Marks: +1, -0

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

\; \frac{1}{x} +\; \frac{1}{y} =

Solution:

Concept:

Use logarithms to express

x

and

y

in terms of a common value, then simplify the sum of reciprocals.

Explanation:

Let

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

(assuming the intended problem uses

6^{z}

, since

6^{2}

would give a constant and options refer to

z

Then

x = \log_{2}k

y = \log_{3}k

, and

z = \log_{6}k

\frac{1}{x} = \log_{k}2

\frac{1}{y} = \log_{k}3

, and

\frac{1}{z} = \log_{k}6

Now

\frac{1}{x} + \frac{1}{y} = \log_{k}2 + \log_{k}3 = \log_{k}6 = \frac{1}{z}

Answer:

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

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