Question

# Find the value of ${\log _3}5 \times {\log _{25}}9.$A. 0B. 1C. 2D. 4

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Hint: We are going to use basic logarithmic formulae to solve the given problem.

Given logarithmic expression is ${\log _3}5 \times {\log _{25}}9.$
Using base change formula ${\log _b}a = \dfrac{{{{\log }_x}a}}{{{{\log }_x}b}}$ , we get
$\Rightarrow {\log _3}5 \times {\log _{25}}9$
$$= \dfrac{{\log 5}}{{\log 3}} \times \dfrac{{\log 9}}{{\log 25}}$$
$= \dfrac{{\log 5}}{{\log 3}} \times \dfrac{{\log {3^2}}}{{\log {5^2}}}$
Using power rule of logarithms ${\log _x}{a^n} = n{\log _n}a$ , to simplify the above term,
$= \dfrac{{\log 5}}{{\log 3}} \times \dfrac{{2\log 3}}{{2\log 5}} = \dfrac{1}{1}$
$$\therefore$$ The value of $${\log _3}5 \times {\log _{25}}9$$ = 1.
Note: Power rule of logarithm is ${\log _x}{a^n} = n{\log _n}a$
Base change formula in logarithms is ${\log _b}a = \dfrac{{{{\log }_x}a}}{{{{\log }_x}b}}$. If no base is written we have to consider that as a natural logarithm with base 10. Or we can consider all logarithms having the same base.