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# If a,b,c are in G.P. (where a,b and c are distinct) . Prove that ${\log _{10}}a, {\log _{10}}b$ and ${\log _{10}}c$ are in AP.

Last updated date: 10th Aug 2024
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Hint: This question is a good example of the combined concept of GP means Geometric Progression and AP means Arithmetic Progression. Common ratio in GP is constant. Similarly, the common difference in AP is constant. With the help of these properties of GP we can derive the required equation for the corresponding AP. As the terms in question are involving logarithms, so some logarithm laws will be applicable here.

It is given that a,b and c are three distinct numbers. These numbers are in GP order.
Now, according to the fundamental property of the GP series of three numbers, square of midterm is equal to the product of first and third terms.
Thus for three terms a,b and c in GP , we have
${b^2} = ac$……(I)
Since we have to get the equation for AP of three terms having log value to the base 10.
Therefore from the eqn (I), after taking log on both side we will get,
${\log _{10}}{b^2} = {\log _{10}}(a \times c)$…..(II)
Now we will apply two laws from the logarithms in the above equation. These laws are,
${\log _{10}}(m \times n) = {\log _{10}}m + {\log _{10}}n$
And
${\log _{10}}{m^n} = n{\log _{10}}m$
Thus in equation II we will have the following result after applying above two laws,
$2{\log _{10}}b = {\log _{10}}a + {\log _{10}}c$
Now doing some simple algebraic transformations we will get,
${\log _{10}}b + {\log _{10}}b = {\log _{10}}a + {\log _{10}}c$
$\Rightarrow$${\log _{10}}b - {\log _{10}}a = {\log _{10}}c - {\log _{10}}b$…..(III)
Now , if three terms m,n and p are in AP then, the common difference between the consecutive terms will be the same. Thus,
$n - m = p - n$
Therefore we can apply the above law in equation III. Thus equation III says that differences of second and first terms are the same as the difference of third and second terms among three terms. This is the property of the AP series.
$\therefore$ ${\log _{10}}a, {\log _{10}}b$ and ${\log _{10}}c$ are in AP.

Note: In such types of questions the fundamental property of GP and AP both are very important. Also, we may get more results by using the fundamental property of HP means Harmonic Progression.