What is log_{1/2}(5/2) ?

The solution to log_{1/2}(5/2) is -log_{2}(5)+1

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simplify log_{1/2}(5/2)

simplify

lo g_{\frac{1}{2}} (\frac{5}{2})

- lo g_{2} (5) + 1

Decimal Notation

- 1.32192 \dots

Solution steps

lo g_{\frac{1}{2}} (\frac{5}{2})

Apply log rule:

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

lo g_{\frac{1}{2}} (\frac{5}{2}) = - lo g_{2} (\frac{5}{2})

= - lo g_{2} (\frac{5}{2})

Apply log rule:

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

lo g_{2} (\frac{5}{2}) = lo g_{2} (5) - lo g_{2} (2)

= - (lo g_{2} (5) - lo g_{2} (2))

lo g_{2} (2) = 1

= - (lo g_{2} (5) - 1)

Apply the distributive law:

- (a - b) = - a + b

- (lo g_{2} (5) - 1) = - lo g_{2} (5) + 1

= - lo g_{2} (5) + 1

lo g_{10} (x - 1) - lo g_{10} (x - 2) - lo g_{10} (5)

- \frac{1}{8} lo g_{2} (\frac{1}{8}) - \frac{7}{8} lo g_{2} (\frac{7}{8})

2 lo g_{10} (m) + 3 lo g_{10} (n)

lo g_{3} (y) + lo g_{3} (y + 2)

\frac{1}{4} lo g_{c} (x) + 5 lo g_{c} (y) - 2 lo g_{c} (x)