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So we know the formula $\ln(ab) = \ln(a) + \ln(b)$, but say I choose $a=-2$ and $b=-1$ we have $\ln(2) = \ln(-1) + \ln(-2)$ which is wrong as $\ln(x)$ only valid for $x>0$. What's wrong with this?

5xum
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Nav Bhatthal
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  • How can one side of the equation be valid and the other be incorrect. – Nav Bhatthal Feb 21 '23 at 13:48
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    This formula is only true for positive $a$ and $b$. – Stef Feb 21 '23 at 13:50
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    If you want to write nonsense, you don't need logarithm. We know the formula $1 = \frac a a$. But say I choose $a = 0$. Then we have $1 = \frac 0 0$, which is wrong since dividing by $0$ doesn't make sense. What's wrong with this? – Stef Feb 21 '23 at 13:51
  • @Stef First time that somebody made me laugh here on mse :D – Falrach Feb 22 '23 at 12:39

1 Answers1

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Every rule is comprised of two parts: its domain and its application. The rule with logarithms is:

If $a$ and $b$ are positive real numbers, then $\ln ab = \ln a + \ln b$.

So you cannot choose $a=-2$, and everything you do after you choose $a=-2$ is irrelevant.

Also, as an aside, I advise you try to separate, in your mind, the concept of a wrong statement from the concept of an undefined statement.

For example, $1 + 1 = 3$ is wrong, but $\frac00=1$ is not so much wrong as it is not really an equation, because the left hand side of the "equation" is not defined.

5xum
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  • Thanks for your answer I've only just discovered logarithms (I'm 16) so I'm getting used to them. – Nav Bhatthal Feb 21 '23 at 14:03
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    A more interesting direction would be changing $\ln(-1) + \ln(-2)$ to $\ln(2)$ in an equation like $x = \ln(-1) + \ln(-2)$. As we started with values outside of the domain of $\ln$, we may come up with solutions to a new equation which were not solutions to the original one. There's a whole host of proofs of $1=0$ which leverage creating fractions with a 0 in the denominator (outside of the domain of division) and then cleverly "canceling" out those zeros to yield a valid equation.... it simply isn't an equivalent equation. – Cort Ammon Feb 21 '23 at 14:10
  • Perhaps "false" would be better than "wrong" when distinguishing between false statements and not-an-equation statements. Presumably, many people would say that 0/0 is definitely wrong even though 1=0/0 is not an equation and thus not a false statement. – Stef Feb 22 '23 at 12:49