Question

If \[{\log _a}x,{\log _b}x,{\log _c}x\] be in H.P., then \[a,b,c\] are in
A. A.P.
B. H.P.
C. G.P.
D. None of these

Answer 1

HINT:
If the \[{\log _a}x,{\log _b}x,{\log _c}x\] in the given question are in Harmonic progression. Because the terms in G.P are in logarithmic form, we use logarithm rules to solve the problem, such as the product rule, quotient rule, and power rule. When each term in GP is divided or multiplied by a non-zero quantity, the new sequence has the same common difference as in G.P.
Formula used:
The centre term of A.P is \[\frac{{(a + b)}}{2}\]
The centre term of G.P is \[\sqrt {ab} \]
The centre term of H.P is \[\frac{{2ab}}{{(a + b)}}\]

Complete step-by-step solution:
The inverse function to exponent is the logarithm. The base\[b\]logarithm of a number is the exponent that must be raised to obtain the number.
We have been given in the question that, \[{\log _a}x,{\log _b}x,{\log _c}x\] are in harmonic progression.
We know that here, Now we have to use base change rule
Now the terms \[\frac{{\log x}}{{\log a}},\frac{{\log x}}{{\log b}},\frac{{\log x}}{{\log c}}\] are in H.P
Then, now we have to write in the term of A.P.
So, we have
The terms \[\frac{{\log a}}{{\log x}},\frac{{\log b}}{{\log x}},\frac{{\log c}}{{\log x}}\] are in Arithmetic progression.
Now we have to use base change rule
The terms \[{\log _x}a,{\log _x}b,{\log _x}c\] are in Arithmetic progression
We have to observe from the above equation that
Now, \[a,b,c\] are in Geometric progression.
Hence, the correct answer is option C.
Note:
Students frequently mix up the nth term and the letter 'n'. Sometimes, they are unable to determine which formula will be used. A geometric progression is a series in which each successive element is derived by multiplying the preceding element by the common ratio, denoted by the letter r. Geometric progression can be either positive or negative. Apply the appropriate formula to the given data and find the unknowns. Be cautious when performing calculations on geometric progression.