Power Log Zero Returns as Zero?

Can someone solve this maths mystery in Matlab? why log(0) returns as -Inf, but 5^log(0) returns as zero?

log(0)

ans =

-Inf

5^log(0)

ans =

0

Answer 1

[John D'Errico]

Um, basic math?

1. The log function is singular at zero. Log(x) approaches -inf, as x approaches zero. So this seems clear.

2. What is 5 raised to some large positive power? A big number.

3. What is 5 raised to some large negative number? It is the inverse of the answer to #2. So a TINY number.

4. What is the smallest number you can represent in double precision without an underflow to zero?

5^-462
ans =
9.8813e-324

5^-463
ans =
     0

realmin
ans =
  2.2251e-308

Realmin for a double is 2.2251e-308, but since you can have what are called denormalized numbers, you can go just a bit further before doubles just give up.

5. Is infinity larger than 463? Last time I checked, yes, it was.

[wesleynotwise]

Gosh. I always thought that anything associated with an infinity will result in infinity. Well, obviously, basic maths is not programmed in my brain.

[Stephen23]

"I always thought that anything associated with an infinity will result in infinity"

What about 1/N as N->inf? You really should complain to your high school that they did a bad job.

[wesleynotwise]

Nah, I blame myself for not paying attention in the class. Bad student. if 1/N and N -> Inf, I would say it is close to zero, But if 1/ Inf, I would say it is infinity.