# Can a logarithm have a negative base?

##### 1 Answer

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^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^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

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

Suppose

What values of

Taking natural logs of both sides:

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

So

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