Logarithm of a negative number?

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.

log5 -125     

log10 -100                    

log5 0                

log4 0

Why the logarithm of a negative number cannot be negative?

Logarithm of a negative number or zero is undefined

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

Recent Articles

  1. Find the Number of Combinations

    Jul 30, 21 06:15 AM

    Learn quickly how to find the number of combinations with this easy to follow lesson.

    Read More

Enjoy this page? Please pay it forward. Here's how...

Would you prefer to share this page with others by linking to it?

  1. Click on the HTML link code below.
  2. Copy and paste it, adding a note of your own, into your blog, a Web page, forums, a blog comment, your Facebook account, or anywhere that someone would find this page valuable.