What is log_{1/6}(4/9) ?

The solution to log_{1/6}(4/9) is -2log_{6}(2)+2log_{6}(3)

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simplify log_{1/6}(4/9)

simplify

lo g_{\frac{1}{6}} (\frac{4}{9})

- 2 lo g_{6} (2) + 2 lo g_{6} (3)

Decimal Notation

0.45258 \dots

Solution steps

lo g_{\frac{1}{6}} (\frac{4}{9})

Apply log rule:

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

lo g_{\frac{1}{6}} (\frac{4}{9}) = - lo g_{6} (\frac{4}{9})

= - lo g_{6} (\frac{4}{9})

Apply log rule:

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

lo g_{6} (\frac{4}{9}) = lo g_{6} (4) - lo g_{6} (9)

= - (lo g_{6} (4) - lo g_{6} (9))

lo g_{6} (4) = 2 lo g_{6} (2)

lo g_{6} (9) = 2 lo g_{6} (3)

= - (2 lo g_{6} (2) - 2 lo g_{6} (3))

Apply the distributive law:

- (a - b) = - a + b

- (2 lo g_{6} (2) - 2 lo g_{6} (3)) = - 2 lo g_{6} (2) + 2 lo g_{6} (3)

= - 2 lo g_{6} (2) + 2 lo g_{6} (3)

- \frac{6}{8} lo g_{2} (\frac{6}{8}) - \frac{2}{8} lo g_{2} (\frac{2}{8})

- lo g_{b} (\frac{1}{5})

lo g_{10} (3) 3

- lo g_{10} (4.4 \cdot 1 0^{- 4})

What is log_{1/6}(4/9) ?
The solution to log_{1/6}(4/9) is -2log_{6}(2)+2log_{6}(3)