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Logarithms

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Question : 41 of 85

Marks: +1, -0

If

\; \frac{1}{\log_{a} x} + \; \frac{1}{\log_{c} x} = \; \frac{2}{\log_{b} x}

a, b, c

are in

G.P.
A.P.
H.P.
None

Solution:

Concept:

Use the change of base formula:

\frac{1}{\log_a x} = \log_x a

.

Explanation:

Given:

\frac{1}{\log_a x} + \frac{1}{\log_c x} = \frac{2}{\log_b x}

Convert each term:

\log_x a + \log_x c = 2 \log_x b

Combine left side:

\log_x (ac) = \log_x (b^2)

Since bases are equal:

ac = b^2

Thus

a, b, c

are in geometric progression (G.P.).

Answer:

G.P.

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