How do you evaluate $log_(1/2) 4$ using the change of base formula?

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Answer 1 · 2017-06-23T19:29:04

$- 2$

We have: $log_(frac(1)(2))(4)$

The change of base formula is $log_(b)(a) = frac(log_(c)(a))(log_(c)(b))$ ; where $c$ is the desired base.

In our case, the base is $frac(1)(2)$ and the argument is $4$ , both of which can be expressed in terms of $2$ .

So let's set $2$ as the new base:

$Rightarrow log_(frac(1)(2))(4) = frac(log_(2)(4))(log_(2)(frac(1)(2)))$

$Rightarrow log_(frac(1)(2))(4) = frac(log_(2)(2^(2)))(log_(2)(2^(- 1)))$

Using the laws of logarithms:

$Rightarrow log_(frac(1)(2))(4) = frac(2 cdot log_(2)(2))(- 1 cdot log_(2)(2))$

$Rightarrow log_(frac(1)(2))(4) = - frac(2 cdot 1)(1)$

$therefore log_(frac(1)(2))(4) = - 2$

How do you evaluate log_(1/2) 4log_(1/2) 4 using the change of base formula?