What is log_{1/2}(2/3) ?

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

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simplify log_{1/2}(2/3)

simplify

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

- 1 + lo g_{2} (3)

Decimal Notation

0.58496 \dots

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

lo g_{2} (2) = 1

= - (1 - lo g_{2} (3))

Apply the distributive law:

- (a - b) = - a + b

- (1 - lo g_{2} (3)) = - 1 + lo g_{2} (3)

= - 1 + lo g_{2} (3)

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