Logarithms

The Basics

To start, let us examine why a logarithm is used in science. One reason to use logarithms is that many things that a scientist would want to quantify are present in vastly different amounts under varying conditions. Concentrations of compounds in solution easily can vary greatly depending upon the conditions of an experiment. For example, the concentration of H₃O⁺ in water tells a chemist the acidity of a solution. In your stomach, which is very acidic, the concentration of H₃O⁺ is 0.01 M; but in household ammonia the concentration of H₃O⁺ is 0.000000000001 M. This corresponds to a 10 billion-fold difference. As we will see later, the use of logarithms will provide a much easier scale to work with this wide range of concentrations.

n^x = Y

then

log_nY = X

So the logarithm of 1000 is 3 because if we placed 3 as the exponent for 10 we get 1000.

Let's try another example:

Problem: log0.01 = ?

0.01 = 10^-2(if you raise 10 to the -2 power, you get 1/10², or 0.01)

log0.01 = -2

So the logarithm of 0.01 is -2 because if we raised 10 to the power of -2 we get 0.01.

These are simple examples because the numbers chosen are exponentials of 10. Notice that the log of such numbers is simply the number of places you would have to move the decimal point to give you 1. This fact will allow you to easily estimate logarithms for more complicated problems. For more difficult numbers, you must turn towards a scientific calculator.

Here are some other helpful properties of logarithms: