Question

How do you evaluate `log_(1/2) 5` using the change of base formula?

Answer 1

I got: `-2.32193`

Explanation:

Consider a log such as:

`log_ba`

we can change into the new base `e` as:

`log_ba=color(red)((log_ea)/(log_eb))`

as you can see the new base gives a fraction between two new logs BUT we can easily use our calculator to evaluate `log_e` that is called the Natural Logarithm and is indicated as `ln`. So we get:

`log_ba=(log_ea)/(log_eb)=ln(a)/ln(b)=`

in our case:

`ln(5)/ln(1/2)=(1.60943)/(-0.69315)=-2.32193`

[I forgot, you can test your result by applying the definition of log:
we got that:
`log_(1/2)(5)=-2.32193`

so:
`(1/2)^(-2.32193)=5`
that can be evaluated with the calculator]