# Comparing apples and oranges

Our story so far:

N_{t} = N_{0} * 2^{t/g}

But we can take this even one step farther. Why is the number "2" in this equation? Only because bacteria multiply by doubling. Not all organisms do that, and "2" as a base doesn't necessarily make sense for them.

In fact, mathematicians like to use a different base: "e". What is e? It is just a number -- a bit less than 3. But its a pretty special number -- you know it has to be, seeing as it has a name.

Remember how we said that with exponential growth: the population always grows by the same factor -- for example, a population might grow by 5% in one minute. But how many actual bacteria are added depends on how big the population already is: a population of 100 bacteria grows by 5, whereas a population of 100 million grows by 5 million. That is, **the actual growth depends on how big the population already is**.

But there is one function where the actual growth doesn't just DEPEND ON the size of the population, it is the SAME AS the size of the population. That function is y = e^{x}.

So for a mathematician, e is a perfect number for talking about growth, just like pi is a perfect number for talking about circles. And growth equations are often expressed as

N_{t} = N_{0} * e ^{mt}

where e is some growth constant.

If you compare the equation just above with the one at the top of the screen, you see that they are very similar. All (!) we need to do is to figure out how

2 ^{1/g} = e ^{m}

This looks kind of daunting at first, but remember that g is whatever the generation time is. And, since we need to get the "m" out of the exponent, we need to use logs (remember: when you have trouble with exponents, use a log). In this case, since the base is "e", we should use ln (natural log) instead of log base 10.

So let's find the value of m for E. nice, which has a generation time of 20 minutes:

2 ^{1/20} = e ^{m}

ln (2 ^{1/20} ) = ln ( e ^{m} )

1/20 * ln(2) = m * ln(e)

m = ln(2)/20 = 0.035

By analogy, for any value of g we would have gotten:

m = ln(2)/g

So here is the really truly last version of this equation:

N_{t} = N_{0} * e ^{0.035 t}, which means the same as

N_{t} = N_{0} * 2 ^{ t/20}, which means the same as

N_{t} = N_{0} * 2 ^{ ts}, where ts = 20 minutes

One more quick calculation for E. fasterus:

m = ln(2)/15 = 0.046

And we can finally compare these two strains on a single graph:

Much better!

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