# Why is pH a negative logarithm?

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• opus
In summary, logarithms are commonly used in pH scales to measure the concentration of a substance, specifically hydronium ions. The negative logarithm is used to ensure that the pH is positive, and this is because the concentration of hydrogen ions is always less than 1. This makes it more convenient for most numbers to be between 0 and 14. However, there are cases where the pH can be negative, and this is not a mathematical necessity but rather a convention for convenience. The pH scale can be adjusted by using different bases or adding a constant, but it does not change the underlying chemistry.f

#### opus

Gold Member
I'm going over applications of logarithms in my College Algebra class and I'm at a part where it talks about pH scales, and it shows the pH concentration of a substance to be the negative logarithm of hydronium ions.
I want to know why the logarithm is negative, so I googled it and the answers all require Chemistry knowledge (obviously). Most answers that I've seen mention having water in the concentration as well and that has something to do with it.

Considering I don't have the Chemistry knowledge (and why I put this thread here instead of the Chemistry forum), is there a way to understand why we are taking the negative logarithm the measure the quantity of a substance?

Actually I think I got it. After plugging some random numbers in, it seems the purpose is the have the pH be positive at the end, although I'm not sure why.
What was tripping me up was the fact that the hydrogen ions were always less than 1, but I suppose that's because it's the concentration of them, not how many there actually are. And since this number will have a negative exponent added to a number between 1 and 9, the result will be negative, which then gets negated again to be positive. Is this the case?

It is just more convenient as most numbers are between 0 and 14. You save a sign most of the time.
There are acids strong enough to create a negative pH. There is nothing special about 1mole/l, as both mole and liters are arbitrary units. Water has about 55 mole per liter, once you have more than 0.02 H3O+ molecule per water molecule the pH gets negative. From a modern point of view the fraction of molecules which are H3O+ would be a more natural choice.

opus
Ok great thank you. So it's not really a mathematical necessity. It's just something that is done for convenience?

You could replace the pH with its negative everywhere without consequences as long as you keep it consistent. You can also add a constant to it. You can also use base e, 2 or whatever you want. This scales the values but it still doesn’t change chemistry.

opus
Understood! Thank you.