Log of Negative Number
Name: Jessica C.
Why can't a log be taken of a negative number?
By definition, the LOGa (N) = L is the power that 'a' must be raised to equal 'N'
that is: a^L = N. Now, you can see the problem that if N < 0 no 'L' exists such
that a number 'a' raised to the power L can equal a negative number. For example
(3)^-2 = 1/(3)^2 > 0 and (-3)^2 = 9 > 0 and (-3)^-2 = 1/(-3)^2 = 1/9 > 0.
Having said that, the same rules do not apply to complex numbers. Some
examples using the definition of LOG's given above you see some
interesting results. Examples:
(-3)^i = 0.01966+0.03849i (3i)^-3 = 0.03704i (-3i)^-3
= -0.03704i (-3i)^3i = -109.996 -17.102i
This is just another example of many that could be cited where a problem that is
impossible, or very difficult using real number algebra become straightforward in
complex number algebra. Which is reason enough to study complex variables.
You can take the log of a negative number, but your answer is no longer "real",
but complex. It has an imaginary part to it. If you have not yet been taught
about complex, imaginary numbers, then this will make no sense.
But for example the natural log of -1 (ln(-1)) is plus or minus i * pi, where i
is the square root of -1, and pi, is your old friend from
circles and diameters.
It all may seem strange, but it works out into a consistent set of rules. The
rules may seem complicated, and useless, but they are not. They show up as
useful shortcuts later, in engineering and science calculations, if nothing
You have asked a tiny little question which leads to a lot more math.
A logarithm is the inverse of a power, just as subtraction is the inverse of
addition. If y=10^x, then x=log(y). Ten raised to a power is always positive,
y is always positive. As a result, you can only take the log of a positive
number. At higher levels of mathematics, "imaginary numbers" are introduced.
Ten raised to an imaginary number can be negative, so the log of a negative
number is imaginary. So long as you use only real numbers that represent real
amounts, the log of a negative number is undefined.
Dr. Ken Mellendorf
Math, Science & Engineering
Illinois Central College
Click here to return to the Mathematics Archives
Update: June 2012