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Popular >

simplify log_{1/9}(sqrt(27))

Solution

simplify

lo g_{\frac{1}{9}} (27)

Solution

- \frac{3}{4}

+1

Solution steps

lo g_{\frac{1}{9}} (27)

Apply log rule:

lo g_{\frac{1}{a}} (x) = - lo g_{a} (x)

lo g_{\frac{1}{9}} (27) = - lo g_{9} (27)

= - lo g_{9} (27)

Factor the number:

9 = 3^{2}

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

Apply log rule:

lo g_{a^{b}} (x) = \frac{1}{b} lo g_{a} (x), a > 0

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

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

27 = 3^{\frac{3}{2}}

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

Apply log rule:

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

lo g_{3} (3^{\frac{3}{2}}) = \frac{3}{2}

= - \frac{1}{2} \cdot \frac{3}{2}

\frac{1}{2} \cdot \frac{3}{2} = \frac{3}{4}

= - \frac{3}{4}

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Frequently Asked Questions (FAQ)

What is log_{1/9}(sqrt(27)) ?
The solution to log_{1/9}(sqrt(27)) is -3/4