# What is a logarithmic function?

Logarithmic functions are a method of condensing a wide range of values. Log 10 functions are often used in biology and other sciences as an easy way to show values that differ by 10-fold orders of magnitude (i.e. 1, 10, 100, 1000, etc.). When we use log 10, we indicate the order of magnitude by placing an exponent on the base number (i.e. 103). The logarithm is the exponent (in this case it's 3). In mathematical terminology we write log functions as:

log10 X = n means 10n= X

## How are logs used in biology?

There are several uses of log functions in biology, most of which deal with the measurement of very small quantities of compounds, particularly ionic compounds. The most widely known is the measurement of pH. pH measures the relative concentration of hydrogen (H+) ions in solution. Because there is a wide concentration that can occur in different solutions, the pH scale goes from 1 to 14. Water, which has roughly the same concentration of H+ and OH- ions and is said to be neutral, has a pH of 7. This means that the H+ concentration in water is 10-7 M (0.0000001M) and the OH- concentration is also 10-7 M. With pH, the H+ and the OH- concentrations add up to 14. The more H+ ions, the more acidic a solution. The fewer H+ ions in solution, the more basic the solution. The logarthmic function for pH is:

pH = -log10[H+]

[context] For example, the fluid in your stomach has a pH of about 4 (lower if you've just drunk a cup of coffee). This means that the fluid has a H+ concentration of 10-4 M and the OH- concentration is 10-10M. On the other hand, your blood has a pH close to 7. This means that your stomach has 1000-fold more H+ in solution than your blood (because if we compare the exponents, there is a difference of 3). This means 103 or 1000.[end context]

Another example is the measurement of calcium ions in a cell. Typically cells have extremely low Ca2+ concentrations, but under certain circumstances, for example, when a muscle cell is stimulated by a nerve, the calcium concentrations can increase exponentially. At rest, the cytosolic concentration of Ca2+ is about 10-14M (0.00000000000001 M; see why we use the exponent notation!!!!), but when a muscle is stimulated, the concentration may rise to 10-4 M due to influx through calcium channels in the cell membrane. The easiest way to explain how much calcium comes into the cell is to say that there is a 1010 increase in Ca2+. This is 1,000,000,000-fold increase in calcium during muscular contraction.

## Examples for you to try:

A solution with a pH of 3 has how many more H+ than a solution with a pH of 6?

1. Turn your pH into exponents.
Show me how to do this.

2. Divide your larger H+ concentration by your smaller H+ concentration. Remember that the bigger the pH, the smaller the actual concentration.
Show me how to do this.

3. Use exponent rules to simplify your expression. If you divide terms, it means to subtract the exponents.
Show me how to do this.