Question

If \[{\log _x}a,{a^{x/2}}\] and \[{\log _b}x\] are in G.P., then \[x = \]
A.\[ \log \left( {{{\log }_b}a} \right)\]
B. \[ {\log _a}\left( {{{\log }_a}b} \right)\]
C. \[{\log _a}\left( {{{\log }_e}a} \right) - {\log _a}\left( {{{\log }_e}b} \right)\]
D. \[{\log _a}\left( {{{\log }_e}a} \right) - {\log _a}\left( {{{\log }_e}b} \right)\]

Answer 1

Hint:
You need to be familiar with the fundamentals of geometric progression in order to answer this question (G.P). In this problem, terms a, b, and c are in the Geometric progression (G.P), which indicates that the square of term b is the sum of terms ‘a’ and ‘c’, or \[{b^2} = ac\]. This Geometric progression reasoning is used with logarithmic formulas to produce the desired outcome.
Formula use:
 Terms a, b, and c are in the Geometric progression (G.P),
 \[{b^2} = ac\]

Complete step-by-step solution
We have been given the question that,
If \[{\log _x}a,{a^{x/2}}\] and \[{\log _b}x\]are in Geometric progression and we have to find the value of \[x\].
The variables a, b, and c are said to be in geometric progression.
We are aware that when variables develop geometrically, the square of the second term equals the sum of the first term and the third term.
The first term in this case is a, the second is b, and the third is c.
Which, indicates \[{b^2} = ac\].
 Therefore, the equation looks like
\[ \Rightarrow {\left( {{{\rm{a}}^{\frac{x}{2}}}} \right)^2} = \left( {{{\log }_b}{\rm{x}}} \right) \times \left( {{{\log }_x}a} \right)\]
Now, it becomes
\[ \Rightarrow {a^x} = \frac{{{{\log }_b}a}}{{{{\log }_b}x}} \times {\log _b}x\]
On simplification by cancelling the similar terms, we obtain
\[ \Rightarrow {{\rm{a}}^x} = {\log _b}a\]
Now let’s apply logarithm to find the desired values.
Taking log on both sides of the above equation, w obtains
\[ \Rightarrow {\rm{x}}\log a = \log \left( {{{\log }_b}a} \right)\]
Thus, we obtain \[ \Rightarrow {\rm{x}} = {\log _a}\left( {{{\log }_b}a} \right)\]
Therefore, if \[{\log _x}a,{a^{x/2}}\] and \[{\log _b}x\] are in G.P., then \[{\rm{x}} = {\log _a}\left( {{{\log }_b}a} \right)\]
Hence, the option A is correct.
Note:
Students are likely to make mistakes in progression type formula as there are so many formulas to remember. In this problem, the mistake mostly occurs while using logarithmic formulas and while dealing with powers. This should be done cautiously to have correct solution.