Why does the base of a log function have to be positive?

Question

Why can't it be negative?

Answer 1

Because the logarithmic function is the inverse function of the exponential function, which is defined only for positive bases (and also the base should not be 1):

log[base a] y = x <-> y = a^x

If a was 1, then
y = 1^x = 1 would be just a horizontal line (which is not an exponential function at all. Same for a = 0.

If a was negative, then the function would not be continuous, neither differentiable, and in general very wild (try sketching a graph of y = (-2)^x and you'll see what I mean.

When you introduce complex numbers, the situation is different, but I guess you're only asking about logs for real numbers.

Hope this helps.

Answer 2

It doesn't have to be. However, using negative bases sometimes gets you non-real answers, and depending on the level of mathematics you study the complex plane might not be taught. This is why you can't have them if you stick with real numbers:

Log(base 2) 8 = 3

since 2^3=8

Log(base -2) 8 =?

Well (-2)^3=-8, and there aren't any other solutions that make sense. Because of the way the sign changes when a negative number is raised to a power, most Logarithms with negative bases wont have solutions.

Also:

Log(base 4) 8 = 1.5

Since 4^(1.5)= 4^1 * 4^.5 = 4 * Sqrt4 = 4*2=8

However

Log(base -4) 8 = ?

-4^1.5= -4^1 * Sqrt-4 and Sqrt-4 is non real, which isn't helpful.

Another way to think of it is using the change of base formula:

Log(base a) b = Ln(b)/Ln(a)

Where Ln is the natural logarithm [who's base is e]. Since Logs of negatives do not have real answers, we avoid this.

However if you deal in the complex plane, all these problems do have solutions. In fact the only logarithm that is not well defined is Log zero.

Answer 3

Because if the base is negative, then the value of the "function" would alternate between positive and negative, and this is discontinuous, and therefore can't be a function.

You can find the log of a number in base -2, but it won't be part of a function.