A logarithm is the number to which a base must be raised to reach another number. e.g. log10(100)=2, so 102 =100. When seen as just log(n) this is because the base is 10. A natural logarithm is a logarithm with a base of the natural constant, which is known as e.
logb(xy) = logb(x) + logb(y)
logb(x/y) =logb(x) - logb(y)
To work out a solution's pH you need to consider the molar concentration of hydrogen ions in solution (that's what acidity is). These concentrations are small in bases and large in acids. The range of hydrogen ion concentrations is incredibly vast (1M to 1 X 10-14M) and so to convert these very small numbers to more manageable ones we use logs to create the pH scale.
The pH of a solution can be worked out by using the equation: -log10 [H+] where H+ is the hydrogen ion concentration in mol dm-3. This can be obtained on all university calculators by using the 'log' function.
Example: the pH of a 0.1M solution of hydrochloric acid has a H+ concentration of 0.1M, so the pH is -log(0.1)= 1. You'll notice this is logical as hydrochloric acid is a strong acid so should have a very low pH.