Solve the following

Question:

If logxaax/2 and logb x are in G.P., then write the value of x.

Solution:

$\log _{x} a, a^{\frac{x}{2}}$ and $\log _{b} x$ are in G.P.

$\therefore\left(\mathrm{a}^{\frac{x}{2}}\right)^{2}=\log _{x} \mathrm{a} \times \log _{\mathrm{b}} x$

$\Rightarrow a^{x}=\frac{\log _{b} a}{\log _{b} x} \times \log _{b} x$

$\Rightarrow a^{x}=\log _{b} a$

Now, by taking $\log _{a}$ on both the sides:

$\Rightarrow \mathrm{x}=\log _{\mathrm{a}}\left(\log _{\mathrm{b}} \mathrm{a}\right)$

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