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Logarithms

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Question : 58 of 85

Marks: +1, -0

If

a^{x}=b^{y}=c^{2}

x, y, z

are in G.P. then

\log a

\log

b

and log c are in

A.P.
G.P.
A.P. and G.P. both
None

Solution:

Concept:

The relationship between

a^x = b^y = c^z

and the progression of

x,y,z

determines the progression of

\log a, \log b, \log c

.

Explanation:

Let

a^x = b^y = c^z = k

(common value).

Then

a = k^{1/x}

,

b = k^{1/y}

,

c = k^{1/z}

.

Take logarithms:

\log a = \frac{1}{x}\log k

,

\log b = \frac{1}{y}\log k

,

\log c = \frac{1}{z}\log k

.

Thus

\log a, \log b, \log c

are proportional to

\frac{1}{x}, \frac{1}{y}, \frac{1}{z}

.

Given

x, y, z

are in GP, so

y^2 = xz

.

Check if

\frac{1}{x}, \frac{1}{y}, \frac{1}{z}

are in GP:

(\frac{1}{y})^2 = \frac{1}{y^2} = \frac{1}{xz} = \frac{1}{x} \cdot \frac{1}{z}

, which holds true.

Hence

\log a, \log b, \log c

are in GP.

Answer:

GP (Geometric Progression).

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