Question

Can a logarithm have a negative base?

Answer 1

Sort of yes, but it's not very useful...

Explanation:

This is a really interesting question.

The answer is basically yes, but it's not generally very useful.

First let's look at logarithms with positive bases.

If `a > 0` and `a != 1`, then:

`a^x: (0, oo)->RR`

is a one-one function with inverse:

`log_a: RR->(0, oo)`

Does this idea extend to negative bases?

If `a < 0` and `n in NN`, then we can quite happily define:

`a^n = stackrel "n times" overbrace ((a)(a)...(a))`

giving us a well defined one to one function:

`a^n:NN->{a^n: n in NN} sub RR`

which has a well defined inverse

`log_a:{a^n: n in NN}->NN`

So in this sense we can say things like `log_(-2) -8 = 3`

Things get more complicated and Complex once we start dealing with fractional exponents.

Suppose `y = a^z` for some `y in RR`

What values of `z` work?

Taking natural logs of both sides:

`ln y = ln a^z = z ln a = z (ln (-a) + i pi (2k+1))`

So `z = ln y / (ln (-a) + i pi (2k+1))` for some `k in ZZ`

That is, we can define:

`log_a(y) = ln y / (ln (-a) + i pi (2k+1))` for some `k in ZZ`

Regardless of what value of `k` we choose, this will have a non-Real Complex value most of the time, but at least we can restrict the domain of `a^z` to get a well defined inverse.