Question

If a, b, c are in G.P., then log a, log b, log c are in?
A) A.P.
B) G.P.
C) H.P.
D) None

Answer 1

Hint: Given a, b, c are in G.P. which means the square of the second term is the product of first term and third term i.e. ${b^2} = ac$ . Apply logarithm to it and solve the logarithm to find which progression are log a, log b, log c in.

Complete step-by-step answer:
We are given that the variables a, b, c are in geometric progression.
We have to find log a, log b, log c are in which progression.
We know that when the numbers or variables are in Geometric Progression, then the square of the 2nd term is equal to the product of the 1st term and 3rd term.
Here a is the 1st term, b is the 2nd term and c is the 3rd term.
This means ${b^2} = ac$
Apply logarithm to the above equation, ${b^2} = ac$
$\log \left( {{b^2}} \right) = \log \left( {ac} \right)$
Where $\log \left( {{a^n}} \right) = n\log a$ , $\log \left( {ab} \right) = \log a + \log b$
$
  \log \left( {{b^2}} \right) = 2\log b \\
  \log \left( {ac} \right) = \log a + \log c \\
  \log \left( {{b^2}} \right) = \log \left( {ac} \right) \\
   \to 2\log b = \log a + \log c \\
 $
When p, q, r are in an Arithmetic progression, then $2q = p + r$
Here
 $
  p = \log a,q = \log b,r = \log c \\
  2q = p + r \\
  2\log b = \log a + \log c \\
 $
Therefore, log a, log b, log c are in Arithmetic Progression.
So, the correct answer is “Option A”.

Note: A Geometric progression is a sequence of numbers in which each term after the first term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. An arithmetic progression is a sequence of numbers in which the difference between any two successive members is a constant, which is called a common difference.