Question

How do you express `3log_5 4 - 5log_5 2` as a single logarithm?

Answer 1

`3 log_5 4 - 5 log_5 2 =1/log_2 5`

Explanation:

To solve this, you will need to use the "Change of Base" formula:
`log_a x = (log_b x) / (log_b a)`
where we change the base-`a` logarithm to base-`b`.

Let's see how we can apply this to what we have.
I notice that we have a 4 and a 2 inside a base-5 logarithm.
So maybe I should change to a base-2 logarithm.
Let's see how each log transforms:
`log_5 4 = (log_2 4)/(log_2 5)`
but since `4=2^2`,
`log_2 4 = log_2 2^2= 2`
so
`log_5 4 = 2/(log_2 5)`
and similarly,
`log_5 2 = 1/(log_2 5)`

So, our expression becomes:
`3 log_5 4 - 5 log_5 2 = (3*2)/(log_2 5) - (5*1)/(log_2 5)`
i.e.
`=(6-5)/(log_2 5)`
`=1/log_2 5`

There could be other answers using different bases.