Question

If three numbers be in G.P., then their logarithms will be in
A. A.P.
B. G.P.
C. H.P.
D. None of these

Answer 1

Hint: Given that a, b, and c are in G.P., the square of the second term is the product of the first and third terms, that is \[{b^2} = ac\]. Apply logarithm to it and solve the logarithm to determine which progression contains log a, log b, and log c.

Complete step by step solution: The variables a, b, and c are shown to be in geometric progression.
We must determine which logs “a”, “b”, and “c” is in which progression.
When the numbers or variables are in Geometric Progression, we know that the square of the second term equals the product of the first and third terms.
The first term is ‘a’, the second term is ‘b’, and the third term is ‘c’
The above statement means that,
\[{b^2} = ac\]
Now we have to apply logarithm to the above equation \[{b^2} = ac\]:
\[\log \left( {{b^2}} \right) = \log \left( {ac} \right)\]
Where:
\[\log \left( {{a^n}} \right) = n\log a\]
\[\log \left( {ab} \right) = \log a + \log b\]
We know that,
\[\log \left( {{b^2}} \right) = 2\log b\]
Now, \[\log \left( {{b^2}} \right)\]can be written as,
\[\log (ac) = \log a + \log c\]
From the above equation, rewrite \[\log (ac)\] as \[\log \left( {{b^2}} \right)\]
Take \[\log \] as common from the above equation, we have
\[\log \left( {{b^2}} \right) = \log (ac)\]
That implies,
\[ \to 2\log b = \log a + \log c\]
When p, q, and r form an arithmetic progression,
\[2q = p + r\]
Here, from the above calculations, we get
\[p = \log a,q = \log b,r = \log c\]--- (1)
We already know that, the arithmetic progression is,
\[2q = p + r\]
Now, substitute the values in the equation (1), we get
\[2\log b = \log a + \log c\]
Therefore, \[\log a\], \[\log b\] and \[\log c\] are in Arithmetic Progression.

Hence, the option ‘A’ is correct.

Note: A geometric progression is a number sequence in which each term following the first is obtained by multiplying the previous term by a fixed, non-zero number known as the common ratio. An arithmetic progression is a number sequence in which the difference between any two successive members is a constant, referred to as the common difference.