This help note is really more for non-mathematicians than non-biologists.
pH may be defined as the negative log to base 10 of the hydrogen ion concentration.
This is a completely accurate and almost completely unhelpful definition that you will find in most textbooks. To make matters worse, pH is really very simple.
Logs to base 10 are an easy way of counting in powers of 10. Thus, log10 of 10 is 1, log10 of 100 is 2, log10 of 1000 is 3 and so on. Count the number of zeros and you know what the log10 is. It works for numbers below 10 as well thus: log10 of 1 is 0, log10 of 0.1 is -1, log10 of 0.01 is -2, log10 of 0.001 is -3 etc. Count the number of figures to the right of the decimal point.
So, the concentration of hydrogen ions ([H+] in a neutral solution is 0.0000001 M. The log10 of 0.0000001 is -7. The negative log10 of 0.0000001 is 7, which is why a neutral solution has a pH of 7. Easy. Everything else is arithmetic.
The important thing to remember is that a solution of pH 6 contains 10 times as many protons as one of pH 7. Saying that the pH of the gastric juice is 1 doesn't sound that impressive until you work out that the proton concentration is more than a million times greater than that of the blood.
A formal description of a log is; "the power to which you must express the base in order to reach the number you first thought of". This is why the log10 of powers of 10 are integers (e.g. log10 of 1000 is 3 because 103 = 1000). The log10 of any number that is not a power of 10 will therefore have a decimal part (called a mantissa). e.g. the log10 of 200 is 2.301 because 102.301 = 200. A pH of 7.4 means that the H+ concentration is 10-7.4 M or 0.0000000398 M