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.
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
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
Jun 06, 23 07:32 AM
May 01, 23 07:00 AM