If ${log}_{30} 3 = a$ and ${log}_{30} 5 = b$ , then the value of ${log}_{30} 75$ is

a + b

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a - b

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2 a + b

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2 b + a

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Solution

The correct option is D $2 b + a$
${log}_{30} (75) = {log}_{30} (3 \times 5^{2})$
Using formula for log multiplication, $log a b = log a + log b$
= ${log}_{30} 3 + {log}_{30} 5^{2} = {log}_{30} 3 + 2 {log}_{30} 5$
Comparing the above with given options,
$∴ {log}_{30} (75) = a + 2 b$
Hence option D is correct.

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