Question

Log1/6(5). I am absolutely clueless as to what i'm supposed to do. The 1/6 is below the log.

Answer 1

Question: Evaluate/death stare at
`log_{\frac{1}6}(5)`.

Let's talk about what logarithms mean.

Suppose
`log_{\frac{1}6}(5)=x`.

That's the same thing as
`5=(\frac{1}6)^x`. It's just been simplified.
(Logarithms are the inverse operations of exponents)

We can use a calculator to evaluate logarithms that are in base 10.
(In this case, the base is 1/6)

How can we change this so that it uses just base 10?
We can use something called the change-of-base formula.

Here's what the change of base formula looks like.


`log_x(n)=(log_y(n))/(log_y(x))`

In this case, we'll set the base
`y` to be 10. (you can set it to whatever you want)
`x` is going to be 1/6, and
`n` is 5.
When the base is 10, we don't have to write it, it's like a plus zero or a times one.


`log_\frac{1}6(5)=\frac{log(5)}{log(\frac{1}6)}`

Punch this into a calculator to find your answer.


`\frac{log(5)}{log(\frac{1}6)} \approx \boxed{-0.8982444017}`

You can always check your answer if you need to, of course.


`(\frac{1}6)^(-0.8982444017) \approx 5`