# If \[{\log _a}x,\,\,{\log _b}x,\,\,{\log _c}x\,\] are in A.P. where x\[ \ne \]1, then show that \[{c^2} = {\left( {ac} \right)^{\left( {{{\log }_a}b} \right)}}\].

Answer

Verified

280.8k+ views

**Hint:**Here in the given question \[{\log _a}x,\,\,{\log _b}x,\,\,{\log _c}x\,\] are in Arithmetic progression, then the middle term will be the arithmetic mean of the other two terms. The terms given in A.P are in logarithmic form so while solving the problem we apply logarithm rules such as product rule, quotient rule and power rule.

**Complete step-by-step solution:**

The logarithm is the inverse function to exponent. The base b logarithm of a number is the exponent that we need to raise in order to get the number.

Given \[{\log _a}x,\,\,{\log _b}x,\,\,{\log _c}x\,\] are in arithmetic progression then the common difference between the consecutive terms will be equal

\[ \Rightarrow \]\[d = \,{\log _b}x - {\log _a}x\]-----------(1)

\[ \Rightarrow d = {\log _c}x\, - {\log _b}x\]-----------(2)

Equating (1) and (2) we get

\[ \Rightarrow {\log _b}x - {\log _a}x = {\log _c}x\, - {\log _b}x\]

On rearranging the terms

\[ \Rightarrow {\log _b}x\, + {\log _b}x = {\log _c}x\, + {\log _a}x\]

\[ \Rightarrow 2{\log _b}x = {\log _a}x + {\log _c}x\]

Using change of base rule, we get

\[ \Rightarrow \]\[2\left( {\dfrac{{\log x}}{{\log b}}} \right) = \dfrac{{\log x}}{{\log a}} + \dfrac{{\log x}}{{\log c}}\]

On taking log x common on both the sides we get

\[ \Rightarrow 2\left( {\log x} \right)\left( {\dfrac{1}{{\log b}}} \right) = \log x\left( {\dfrac{1}{{\log a}} + \dfrac{1}{{\log c}}} \right)\]

Cancelling log x on both the sides the equation reduces as following

\[ \Rightarrow 2\left( {\dfrac{1}{{\log b}}} \right) = \left( {\dfrac{1}{{\log a}} + \dfrac{1}{{\log c}}} \right)\]-------(3)

On simplifying equation (3) we get

\[ \Rightarrow \dfrac{2}{{\log b}} = \dfrac{{\log c + \log a}}{{\log a\log c}}\]-------(4)

On rearranging the equation (4)

\[ \Rightarrow 2\log c = \left( {\dfrac{{\log b}}{{\log a}}} \right)\left( {\log c + \log a} \right)\,\,\,\,\]--------(5)

Using product rule \[\log m + \log n = \log mn\] in equation (5)

\[ \Rightarrow 2\log c = \left( {{{\log }_a}b} \right)\left( {\log ac} \right)\]-------(6)

Using power rule \[n\log m = \log {m^n}\] in equation (6) we get

\[ \Rightarrow \log {c^2} = \log {\left( {ac} \right)^{\left( {{{\log }_a}b} \right)}}\]-------(7)

Now cancelling log on both the sides we get

\[ \Rightarrow {c^2} = {\left( {ac} \right)^{\left( {{{\log }_a}b} \right)}}\]

Hence proved

**Note:**Arithmetic progression is a sequence of numbers in which each successive term is a sum of its preceding term and a fixed term. We use \[{T_n} = a + \left( {n - 1} \right)d\]to find any term \[{T_n}\] in the given arithmetic progression, where ‘a’ is the first term of the A.P, nth position of the term, d is the common difference. Similarly \[{S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)\] is used to find the sum of n terms of the A.P.

Recently Updated Pages

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Find the values of other five trigonometric ratios class 10 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Which one of the following places is unlikely to be class 8 physics CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

What is the past tense of read class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Elucidate the structure of fructose class 12 chemistry CBSE

What is pollution? How many types of pollution? Define it