Exponential growth (doubling time, generation time)
In microbiology, populations often grow approximately exponentially during the log (exponential) phase.
This calculator uses the ideal exponential model, which assumes a constant growth rate and no resource limitation
over the time interval of interest.
Assumptions (when this model is valid)
The calculations are most appropriate when cells are in exponential phase: nutrients are not limiting, waste products do not
strongly inhibit growth, and the per-capita growth rate is approximately constant. If the culture begins to approach stationary
phase, exponential predictions can overestimate the final population.
Key quantities
Use N0 for the initial amount (cells, CFU, or any consistent unit), N(t) for the amount after time t,
td for the doubling time (also called generation time), and g for the number of doublings (generations).
Base-2 “doublings” model
If the population doubles every td units of time, then the number of generations is:
\[
g = \frac{t}{t_d}
\]
The population after time t is then:
\[
N(t) = N_0 \cdot 2^{g} = N_0 \cdot 2^{t/t_d}
\]
Solving common lab questions (base-2 form)
1) Solve for final population N(t) (given N0, t, td)
\[
N(t) = N_0 \cdot 2^{t/t_d}
\]
2) Solve for time t (given N0, N, td)
\[
\begin{aligned}
\frac{N}{N_0} &= 2^{t/t_d} \\
\log_2\!\left(\frac{N}{N_0}\right) &= \frac{t}{t_d} \\
t &= t_d \cdot \log_2\!\left(\frac{N}{N_0}\right)
\end{aligned}
\]
3) Solve for doubling time td (given N0, N, t)
\[
\begin{aligned}
g &= \log_2\!\left(\frac{N}{N_0}\right) \\
t_d &= \frac{t}{g} = \frac{t}{\log_2(N/N_0)}
\end{aligned}
\]
4) Solve for generations g (given N0, N)
\[
g = \log_2\!\left(\frac{N}{N_0}\right)
\]
Base-e exponential model (growth rate μ)
Another common form uses the natural exponential function:
\[
N(t) = N_0 \cdot e^{\mu t}
\]
where μ is the specific growth rate (for example, in h-1 if t is in hours). Solving for μ gives:
\[
\mu = \frac{\ln(N/N_0)}{t}
\]
Connecting μ and doubling time
The base-2 and base-e forms describe the same process. One doubling corresponds to multiplying by 2, and
\(2 = e^{\ln 2}\). This leads to:
\[
\begin{aligned}
2 &= e^{\ln 2} \\
N_0 \cdot 2^{t/t_d} &= N_0 \cdot e^{(\ln 2)\,t/t_d} \\
\Rightarrow\ \mu &= \frac{\ln 2}{t_d} \\
\Rightarrow\ t_d &= \frac{\ln 2}{\mu}
\end{aligned}
\]
Why semi-log plots become straight lines
In exponential growth, taking a logarithm converts the curve into a line. For the base-e model:
\[
\begin{aligned}
N(t) &= N_0 \cdot e^{\mu t} \\
\ln N(t) &= \ln N_0 + \mu t
\end{aligned}
\]
That is why plotting log(N) versus time yields a straight line during log phase, and the slope is related to μ.
Timeline table meaning
When the calculator outputs a timeline table, each row is computed from the same model:
\(N(t) = N_{0}\cdot 2^{t/t_{d}}\) (or equivalently \(N(t) = N_{0}\cdot e^{\mu t}\)).
The table is useful for planning sampling times or visualizing how quickly populations increase under exponential growth.
