We will show that the logarithm of a negative number or zero is undefined or does not exist.

Similarly, you cannot find the logarithm of the following expressions.

log_{5} -125

log_{10} -100

log_{5} 0

log_{4} 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 = b^{x}, then log_{b} y = x

If x is bigger than zero or x is equal to zero, It is obvious that b^{x} 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 = b^{x}

65^{0} = 1

7^{0} = 1

24^{1} = 24

2^{3} = 8

4^{2} = 16

5^{3} = 125

6^{4} =1296

8^{5} = 32768

How about when x is negative?

Let x = -2, -3, and -8 and let b = 5

y = 5^{-2} = 1 / 5^{2} = 1 / 25 = 0.04

y = 5^{-3} = 1 / 5^{3} = 1 / 125 = 0.008

y = 5^{-8} = 1 / 5^{8} = 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

log_{b} y with zero or a negative number.

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

log_{5} -125 log_{10} -100 log_{5} 0 log_{4} 0

In fact, for log_{5} -125, there is no number x, such 5^{x} = -125

If you choose 3, you will get 5^{3} = 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 log_{5} 0, there is no number x such that 5^{x} = 0