Question

Write -5 as a logarithim with a base of 4

Answer 1

`log_4((1)/(4^5)) = -5`

Explanation:

`log_b(a) = c` ↔
`b^c = a`

You are given
`b` = 4

You re also given
`c` = -5

To find a, use the exponential form:

`b^c = a`

`= (4)^(-5) = (1)/(4^5) = a`

Then you can convert into a logarithm.

`log_4((1)/(4^5)) = -5`

Answer 2

There are multiple ways of writing this:

`\sf -5log_4(4)=log_4(4^(-5))=log_4\left((1)/(4^5)\right)=log_4\left((1)/(1024)\right)`

Explanation:

Log rule:
`\sf log_a(a)=1`

`\sf \implies log_4(4)=1`

`\sf \implies -5log_4(4)=-5`

Log rule:
`\sf c \cdot log_ab=log_a(b^c)`

`\sf \implies -5log_4(4)=log_4(4^(-5))`

`\sf = log_4\left((1)/(4^5)\right)`

`\sf = log_4\left((1)/(1024)\right)`