The bit about the mantissa being positive is an old convenience for
computation. In the days when people did math by hand or slide rule they would
often multiply strings of numbers by adding the logarithms or exponentiate by
multiplying logarithms. This is how a slide rule works. A convenient way of
representing numbers was in mantissa-exponent notation, or "scientific"
notation. Example 50000 would be 5 times 10 to the 4th power.
When you look up a number in a log table you look up the mantissa or 5. The
log of 5 is about .70 so the log of 5 times 10 to the 4th would be 4.70. The
log of 500 is 2.70 and of 50 is 1.70.
Lets keep dividing by 10. The log of 5 is 0.70 and the log of 0.5 is -0.30.
What happened to the .70? In going from positive to negative subtracting one
no longer gives the expected number ending in .70. This is why you might see
the log of 0.5 in the old tables as "9.70 - 10"
If you combined the positive mantissa for log of 5 (0.7) with the negative
exponent (-1) the result is -0.3.
-----Original Message-----
From: Marshall Dudley <[email protected]>
To: [email protected] <[email protected]>
Date: Thursday, November 11, 1999 2:15 PM
Subject: Re: CS an PH concerns
Bill Schramm wrote:
I am sorry, that is not correct. Check your math. Negative numbers
do not have logarithms, but logarithms THEMSELVES can be negative, and in fact
are negative for any (positive) number between 0 and 1. The log of 0.1 is
minus 1, the log of 0.01 is minus 2, etc. The pH is actually defined as MINUS
the logarithm of the hydrogen ion concentration in moles per liter. So a pH of
7 means that the logarithm of the hydrogen ion concentration is MINUS 7, or
the concentration itself is ten to the minus seventh power (moles per liter).
Only in extremely strong, concentrated acids will there be more than one mole
per liter. When this is true, the logarithm of the molar concentration will be
positive and therefore the pH will by definition be negative. The bottle of
muriatic acid says 20% concentration. This would mean that in a liter of acid
(1000 grams) about 200 grams would be HCl. Divide this by the molecular
weight of HCl which is about 36 and you get a little over 5 moles per liter
which would correspond to a pH of minus 0.7 If you want a more reasonable
approximation of stomach acid look up the pH of stomach acid and dilute
accordingly. Example, (I'm not a biologist) if stomach acid were pH 2, you
should dilute the muriatic acid 500 to 1.
Sorry, you are right. I remember being taught that for logarithms of
numbers below 1, that the exponent part is negative, but the mantissa is always
positive, something I always considered very strange.
Marshall
