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