What is log_{4}(log_{9}(81)) ?

The solution to log_{4}(log_{9}(81)) is 1/2

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simplify log_{4}(log_{9}(81))

simplify

lo g_{4} (lo g_{9} (81))

\frac{1}{2}

Decimal Notation

0.5

Solution steps

lo g_{4} (lo g_{9} (81))

Factor the number:

4 = 2^{2}

= lo g_{2^{2}} (lo g_{9} (81))

Apply log rule:

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

lo g_{2^{2}} (lo g_{9} (81)) = \frac{1}{2} lo g_{2} (lo g_{9} (81))

= \frac{1}{2} lo g_{2} (lo g_{9} (81))

lo g_{9} (81) = 2

= \frac{1}{2} lo g_{2} (2)

Apply log rule:

lo g_{a} (a) = 1

lo g_{2} (2) = 1

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

Multiply the numbers:

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

= \frac{1}{2}

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