Question 50

If $$\log_{x} a,  a^{\frac{x}{2}} and  \log b  x$$ are in GP then $$x$$ is

Solution

For three terms $$A,B,C$$ to be in GP, it must follow : $$B^2=AC$$

Now, $$\log_{x} a, a^{\frac{x}{2}}$$ and $$\log_b x$$ are in GP

=> $$(a^{\frac{x}{2}})^2=(\frac{\log a}{\log x})\times(\frac{\log x}{\log b})$$

=> $$a^x=\frac{\log a}{\log b}$$

=> $$a^x=\log_b a$$

=> $$x=\log_a(\log_b a)$$

=> Ans - (A)


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