Since the logarithm is defined as the inverse function of the exponential function, the domain of log x is exactly the range of e^x, i.e. RR^+.

Jacoby Erickson

Jacoby Erickson

Answered question

2022-10-29

Is there proof show that log x is undefined and make no sense at x = 0?
Is there proof show that log x is undefined at x = 0?
Note(01):: log is the inverse function of the exponential function.
note(02): I edited my question as I meant why it's not make a sens at x = 0?
Thank you for your help .

Answer & Explanation

inmholtau5

inmholtau5

Beginner2022-10-30Added 16 answers

We could define log 0 in whatever way we like, but a sensible definition should preserve the main property of the logarithm, that is,
log ( x y ) = log x + log y
Suppose we set log 0 = a; then, taking y = 0 in the formula above, we have
log ( x 0 ) = log x + log 0
that is,
a = log x + a
and we conclude that log x = 0. But x can be any positive number! So defining the logarithm at 0 to be some real number, forces log x = 0 for any other x > 0
Not really a useful function, I believe you can agree, and certainly not the inverse to the exponential function.

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