What is log_{3}(27x^2) ?

The solution to log_{3}(27x^2) is 3+2log_{3}(x)

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simplify log_{3}(27x^2)

simplify

lo g_{3} (27 x^{2})

3 + 2 lo g_{3} (x)

Solution steps

lo g_{3} (27 x^{2})

Factor the number:

27 = 3^{3}

= lo g_{3} (3^{3} x^{2})

Apply log rule:

lo g_{a} (x y) = lo g_{a} (x) + lo g_{a} (y)

lo g_{3} (3^{3} x^{2}) = lo g_{3} (3^{3}) + lo g_{3} (x^{2})

= lo g_{3} (3^{3}) + lo g_{3} (x^{2})

Apply log rule:

lo g_{a} (a^{b}) = b

lo g_{3} (3^{3}) = 3

= 3 + lo g_{3} (x^{2})

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x), assuming x > 0

lo g_{3} (x^{2}) = 2 lo g_{3} (x)

= 3 + 2 lo g_{3} (x)

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