What is log_{3}(1/(27x^2)) ?

The solution to log_{3}(1/(27x^2)) is -2log_{3}(x)-3

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simplify log_{3}(1/(27x^2))

simplify

lo g_{3} (\frac{1}{27 x ^{2}})

- 2 lo g_{3} (x) - 3

Solution steps

lo g_{3} (\frac{1}{27 x ^{2}})

Apply log rule:

lo g_{c} (\frac{a}{b}) = lo g_{c} (a) - lo g_{c} (b)

lo g_{3} (\frac{1}{27 x ^{2}}) = lo g_{3} (1) - lo g_{3} (27 x^{2})

= lo g_{3} (1) - lo g_{3} (27 x^{2})

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

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

= lo g_{3} (1) - (lo g_{3} (x^{2}) + lo g_{3} (27))

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

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

= lo g_{3} (1) - (lo g_{3} (27) + 2 lo g_{3} (x))

Apply log rule:

lo g_{a} (1) = 0

= 0 - (2 lo g_{3} (x) + lo g_{3} (27))

lo g_{3} (27) = 3

= 0 - (2 lo g_{3} (x) + 3)

0 - (3 + 2 lo g_{3} (x)) = - (3 + 2 lo g_{3} (x))

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

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

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

- 2 ln (\frac{1}{5})

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What is log_{3}(1/(27x^2)) ?
The solution to log_{3}(1/(27x^2)) is -2log_{3}(x)-3