Can the log of 5 equal negative 1?

Answer 1

Having given no base to your log and not specified natural or common, I assume you are asking if there is a base in which logyb5 = -1, ie is there is a number b such that b-1 = 5?

Yes.

If you want to know what that base is, then you can work it out:

Using:

logbx = logax ÷ logab

you can work out the value for b using, say natural logs (base e written loge or usually abbreviated to ln) since we know x=5 and logb5 = -1:

-1 = ln 5 ÷ ln b

⇒ ln b = -ln 5

⇒ b = e-ln 5

⇒ b = 1÷ eln 5

⇒ b = 1/5

Alternatively you can see it straight away:

b-1 = 1/b = 5

⇒ b = 1/5 = 0.2

ie log0.25 = -1