# Logarithm of a negative number?

We will show that the logarithm of a negative number or zero is undefined or does not exist. For example, you cannot find the logarithm of the following expressions.

log5 -125                       log10 -100                     log5 0                 log4 0

The reason for this is that any positive number b raised to any power x cannot equal to a number y less than or equal to zero.

The definition says If y = bx, then logb y = x

If x is bigger than zero or x is equal to zero, It is obvious that bx will be bigger than zero as you can see in the examples below. As a result, y will also be bigger than zero since y = bx

650 = 1

70  = 1

241 = 24

23 = 8

42 = 16

53 = 125

64  =1296

85 = 32768

How about when x is negative?

Let x  = -2, -3, and -8 and let b = 5

y = 5-2 = 1 / 52 = 1 / 25 = 0.04

y = 5-3 = 1 / 53 = 1 / 125 = 0.008

y = 5-8 = 1 / 58 = 1 / 390625 = 0.00000256

As you can see, although y can get very small or very close to zero, it will never be equal to zero or worse be a negative number. That is the key concept here!

Since y can never be zero or negative, it does not make sense to replace y in
logb y with zero or a negative number.

Now, you can clearly see why these expressions do not make sense

log5 -125                       log10 -100                     log5 0                 log4 0

In fact, for log5 -125, there is no number x, such 5x = -125

If you choose 3, you will get 53 = 125 and if you choose -3, you will get 5-3 = 0.008

If it did not work for x = 3 and x = -3, no other numbers will work!

By the same token, for log5 0, there is no number x such that 5x = 0

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