Question

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If a,b,c are in G. P., Prove that:

${\sf {log_{ a}}}{\sf {x,_{}}}{\sf {log_{ b}}}{\sf {x,_{}}}{\sf {log_{ c}}}{\sf {x_{}}} \sf \: are \: in \: H.P$
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Answer 1

$\large\underline{\text{Important concepts}}$

$\red{\bigstar}$Three numbers $a,b,c$ in each type of sequence.

$\implies\boxed{a,b,c\text{ (A.P)}\iff b=\dfrac{a+c}{2}}$

$\implies\boxed{a,b,c\text{ (G.P)}\iff b^{2}=ac}$

$\implies\boxed{a,b,c\text{ (H.P)}\iff b=\dfrac{2ac}{a+c}}$

$\red{\bigstar}$Properties of logarithms.

$\implies\log ab=\log a+\log b$

$\implies\log m^{n}=n\log m$

$\implies\log_{a}b=\dfrac{\log b}{\log a}$

$\large\underline{\text{Solution}}$

Since $a,b,c$ are in G.P

$\implies b^{2}=ac$

Taking log on both sides

$\implies\log b^{2}=\log ac$

By the first and second properties of logarithms

$\implies 2\log b=\log a+\log c$

So

$\implies \log b=\dfrac{\log a+\log c}{2}$

Hence

$\implies\log a,\log b,\log c\text{ (A.P)}$

Dividing by $\log x$

$\implies\dfrac{\log a}{\log x},\dfrac{\log b}{\log x},\dfrac{\log c}{\log x}\text{ (A.P)}$

According to the definition of H.P

$\implies\dfrac{\log x}{\log a},\dfrac{\log x}{\log b},\dfrac{\log x}{\log c}\text{ (H.P)}$

By the third property of logarithms

$\implies\log_{a}x,\log_{b}x,\log_{c}x\text{ (H.P)}$

Hence proven.

$\large\underline{\text{Proof of the concept No.1}}$

The common difference in an A.P is equal.

$\implies d\text{(common difference)}$

$\implies d=a-b,d=b-c$

$\implies a-b=b-c$

$\implies 2b=a+c$

$\implies b=\dfrac{a+c}{2}\text{ (Arithmetic mean)}$

The common ratio in a G.P is equal.

$\implies r\text{(common ratio)}$

$\implies r=\dfrac{a}{b},r=\dfrac{b}{c}$

$\implies \dfrac{a}{b}=\dfrac{b}{c}$

$\implies b^{2}=ac\text{ (Geometric mean)}$

An H.P consists of reciprocals of an A.P.

$\implies\dfrac{1}{a},\dfrac{1}{b},\dfrac{1}{c}\text{ (A.P)}$

$\implies \dfrac{2}{b}=\dfrac{1}{a}+\dfrac{1}{c}$

$\implies \dfrac{1}{b}=\dfrac{a+c}{2ac}$

$\implies b=\dfrac{2ac}{a+c}\text{ (Harmonic mean)}$

$\large\underline{\text{Additional information}}$

We can notice that each result is the A.M, G.M, and H.M. Thereby, we can use the concept of means for finding a term between the two.

Answer 2

Answer:

$\large \dag$ Question provided with us :-

If a,b,c are in G.P ,Prove that ,

${\sf {log_{ a}}}{\sf {x,_{}}}{\sf {log_{ b}}}{\ {x;_{}}}{\sf {log_{ c}}}{\sf {x_{}}} \: are \:in \:HP$

$\large \dag$ We should need to find ?

Here,we should need to need to prove the given value .

$\large \dag$ Solutions :-

• Hey mate,here refer the above given attachment for more information.

$\large \dag$ Some Hints :-

• Here,According to the given question first
• ${b}^{2} = ac$
• taking log on both sides

• Next,we should know all the logarithms properties. According to that we apply here

• After this we will get some value then by dividing log x

• After ,this according to the third property of logarithm we will get the what we should prove which is given in question.

Therefore,

• This is the perfect answer to your question.