Simplify the given equation $${\log _b}x \cdot {\log _a}b$$
A.${\log _x}a$
B.${\log _x}b$
C.${\log _a}x$
D.${\log _b}x$

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Hint: We are going to simplify the given expression using basic logarithmic formulae.
Given expression is $${\log _b}x \cdot {\log _a}b$$
Using the change of base rule from logarithms, we can write the above expression as
$ = \frac{{\log x}}{{\log b}} \times \frac{{\log b}}{{\log a}}$
On simplifying this we get
$ = \frac{{\log x}}{{\log a}}$
$ = {\log _a}x$
We simplified the given expression using change of base rule of logarithms. That is
${\log _b}a = \frac{{{{\log }_x}a}}{{{{\log }_x}b}}$
This formula allows us to rewrite the logarithm in terms of logarithms written in another base.
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