Discrete Mathematics--general question?

Question

I have a T/F question which says, "Justify your answer. For all positive real numbers x, log(x) > 0

How would I go about finding "all" instances of this, or is it generally assumed that finding *any* single instance of proof/disproof is enough? I already know that if x=1 this statement is false. Log(1) = 0 and 0 is not > 0. Is that enough?

I'm studying this alone in my own free time so any additional advice for studying this type of math is appreciated.

Answer 1

Hi,

You are correct that this is false. One contradiction like log(1) = 0 is sufficient to prove it is not always true. For what it is worth, realize that the log of any number between 0 and 1 is a negative value.

log (1/100) = log(1/10²) = log(10^(-2)) = -2

I hope that helps!! :-)

Answer 2

General principle: One counterexample to a
statement is enough to prove it false.
So log(1) = 0 disproves the statement.

Answer 3

You got it, Matt. That is "disproof by example". Keep up the good work and keep posting questions here if you have a problem.

Answer 4

You're correct. To prove it false you only have to give one counterexample and using x=1 is a particularly obvious one.

Answer 5

so.. itll be false.. for all positive real numbers GREATER THAN 1, log (x) > 0