### Chapter 24: Mathematics

Which of the following is an example of a logarithmic scale?

**(A)** Likert scale

**(B)** pH scale

**(C)** visual analog scale

**(D)** verbal rating scale

**(E)** centigrade temperature scale

**The answer is B.** The logarithm of a given number is the exponent to which a fixed base number must be raised to produce that number. As an example, the logarithm of 10,000 to base 10 is 4, since 10^{4} is 10,000. Logarithmic scales are common in physiology and medicine. Half-lives of drugs or radioactivity are measured using logarithmic scales, as are decibels and pH. They are a useful way of presenting information graphically on a single page that would otherwise be difficult due to the large range of values.

pH is a measure of the concentration of hydrogen ions in aqueous solution. It is defined as the negative logarithm of that concentration and mathematically can be written as:

pH = -log_{10} [H+]

Since the scale is logarithmic, every unit change increases or decreases the concentration by a factor of 10. The number of hydrogen ions in solution at a pH of 6 is 10 times that in a solution with a pH of 7.

The visual analog, verbal rating, Likert, and centigrade scales are all linear scales, where any integer increase in value is equal in intensity or degree of change to any other similar integer increase.

**Ref:** Barrett KE, Boitano S, Barman SM, et al. *Ganong's Review of Medical Physiology*, 24th ed. New York, NY: McGraw Hill; 2012.

A resident puts a sandwich contaminated with *S. aureus* in her bag at 9 a.m. in the morning and keeps it at room temperature until noon. Assuming there were 200 colony-forming units (CFU)/mL at 9 a.m. and the bacteria divide every 20 minutes at room temperature, what will the population density of bacteria be when she takes a big juicy bite of it at noon?

**(A)** 2^{9} CFU/mL

**(B)** 51,200 CFU/mL

**(C)** 102,400 CFU/mL

**(D)** 200^{9} CFU/mL

**(E)** too gross to bother answering

**The answer is C.** This problem involves exponential growth (in this case, of bacteria). In general, we describe exponential growth in biological systems in terms of doubling times, or the time it takes for the population to increase ...