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Popular >

simplify log_{1/9}(27)

Solution

simplify

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

Solution

- \frac{3}{2}

+1

Decimal Notation

- 1.5

Solution steps

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

Apply log rule:

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

= - lo g_{9} (27)

Rewrite

9

as

3^{2}

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

Apply log rule:

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

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

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

Rewrite

27

in power-base form:

27 = 3^{3}

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

Apply log rule:

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

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

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

Apply log rule:

lo g_{a} (a) = 1

lo g_{3} (3) = 1

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

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

= - \frac{3}{2}

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Frequently Asked Questions (FAQ)

What is log_{1/9}(27) ?
The solution to log_{1/9}(27) is -3/2