Question

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

Answer 1

Hint: Use the formula of G.P and take log on both sides.

Complete step by step solution: To find the series of \[{\log _a}x,{\log _b}x,{\log _c}x\]
The given equation is
\[ = > {b^2} = ac\]
Substitute the values on the formula
\[ = > 2\log b = \log a + \log c\]
Divide by \[logx\] on both the sides
\[ = > 2\dfrac{{\log b}}{{\log x}} = \dfrac{{\log a}}{{\log x}} + \dfrac{{\log c}}{{\log x}}\]
By solving the equation, it becomes
\[ = > 2{\log _x}b = {\log _x}a + {\log _x}c\]
\[ = > \dfrac{2}{{{{\log }_b}x}} = \dfrac{1}{{{{\log }_a}x}} + \dfrac{1}{{{{\log }_c}x}}\]
Hence, the terms of this equation are in H.P.
So, the terms \[{\log _a}x,{\log _b}x,{\log _c}x\] are in H.P.

Therefore, the correct option is C.

Additional Information: A base should be raised to a particular exponent or power, or index, so as to supply a particular range. The opposite technique to put in writing exponents in arithmetic is victimization logarithms. A range's base-based index is up to another number. An index performs exponentiation's actual opposite performance. An index to base e is stated as Ln. This additionally goes by the name logarithm. A Napierian logarithm is another name for this.
To determine the worth of the exponent perform, utilize the index table. Victimization of the log table is the simplest methodology for determining the worth of the given exponent. That no matter the exponent base, the index of one is usually up to zero.


Note: The logarithm is exponentiation's opposite function in mathematics. This indicates that the exponent to which a fixed number, base b, must be raised in order to create a specific number x, is represented by the logarithm of that number.
To determine what exponents to multiply into a given number, use logarithms.