
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
233.1k+ views
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.
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.
Recently Updated Pages
JEE Main 2023 April 6 Shift 1 Question Paper with Answer Key

JEE Main 2023 April 6 Shift 2 Question Paper with Answer Key

JEE Main 2023 (January 31 Evening Shift) Question Paper with Solutions [PDF]

JEE Main 2023 January 30 Shift 2 Question Paper with Answer Key

JEE Main 2023 January 25 Shift 1 Question Paper with Answer Key

JEE Main 2023 January 24 Shift 2 Question Paper with Answer Key

Trending doubts
JEE Main 2026: Session 2 Registration Open, City Intimation Slip, Exam Dates, Syllabus & Eligibility

JEE Main 2026 Application Login: Direct Link, Registration, Form Fill, and Steps

JEE Main Marking Scheme 2026- Paper-Wise Marks Distribution and Negative Marking Details

Understanding the Angle of Deviation in a Prism

Hybridisation in Chemistry – Concept, Types & Applications

How to Convert a Galvanometer into an Ammeter or Voltmeter

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

NCERT Solutions For Class 11 Maths Chapter 12 Limits and Derivatives (2025-26)

NCERT Solutions For Class 11 Maths Chapter 10 Conic Sections (2025-26)

Understanding the Electric Field of a Uniformly Charged Ring

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

Derivation of Equation of Trajectory Explained for Students

