# Simplify the given equation $${\log _b}x \cdot {\log _a}b$$

A.${\log _x}a$

B.${\log _x}b$

C.${\log _a}x$

D.${\log _b}x$

Last updated date: 24th Mar 2023

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Answer

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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$

Note:

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.

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$

Note:

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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