What is log_{8}(sqrt(2)) ?

The solution to log_{8}(sqrt(2)) is 1/6

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simplify log_{8}(sqrt(2))

simplify

lo g_{8} (2)

\frac{1}{6}

Decimal Notation

0.16666 \dots

Solution steps

lo g_{8} (2)

Factor the number:

8 = 2^{3}

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

Apply log rule:

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

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

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

Apply radical rule:

a = a^{\frac{1}{2}}

2 = 2^{\frac{1}{2}}

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

Apply log rule:

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

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

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

\frac{1}{3} \cdot \frac{1}{2} = \frac{1}{6}

= \frac{1}{6}

\frac{13}{39}

\frac{\frac{8 r + 8 s}{r ^{2}}}{\frac{r ^{2} - s ^{2}}{8 r}}

(x y^{2} - 3 x^{2} - y^{2} + x^{2} y) + (x^{2} y + 5 x^{2}) + (3 x y^{2} - y^{2} - 5 x^{2})

\frac{- 8}{1}

\frac{6}{7} \div 4