# 1: In search of … the exact doubling time

## Learning Outcomes

When you have completed this module, you should be able to:

• Use the growth equation to calculate generation time from a set of data
• Know how to use logarithms (logs) to solve equations with exponents

## In search of … the exact doubling time

So far, we’ve been the doubling or generation time from a graph of the cell numbers over time. But there is a problem… Guesstimating a parameter from a graph is not really accurate scientific procedure. It would be hard to imagine publishing a report in an academic journal that started out “we looked at the graph, squinted a little, and decided that the doubling time was 23 minutes…”.

It is possible to look up the doubling time in a table of data, but only if the table happens to include an exactly doubled population. If the table does NOT contain a doubled population entry, then you’re in trouble. In other words, we can figure out the exact doubling time for the time series with convenient numbers below, but not for the one with any general sort of number:

Time convenient general
10:00 am 10 million 10 million
10:20 am 20 million 15 million

Clearly, the convenient example doubled in 20 minutes. And at first glance, it appears that the more general example got “halfway” to doubling, so doubling time should be 40 minutes. Let’s see if that’s true — if the population keeps growing at the same rate, will it double in 40 minutes?

time general
10:00 10 million
… multiply by 1.5 to get
10:20 15 million
… multiply by 1.5 to get
10:40 22.5 million

Oops, we got to 22.5, not 20 million. Granted, this is a small difference, but it’s still not the right answer. As the old joke goes, a million here, a million there, pretty soon you’re talking about a real epidemic. What we need is a foolproof way to determine exactly what the doubling time is.