Log-phase growth rate (μ) calculator
During the log (exponential) phase of microbial growth, population size increases approximately exponentially.
In this phase, a constant specific growth rate μ (mu) provides a compact description of how fast the population grows
per unit time.
Model used in this calculator
The calculator assumes ideal exponential growth over the chosen time window:
N(t) = N0 · eμt.
Taking the natural logarithm gives a straight-line relationship:
ln(N(t)) = ln(N0) + μt.
In other words, μ is the slope of the line on a semi-log plot of ln(N) versus time.
Why the calculator warns “use only log-phase”
Real cultures often include a lag phase (slow start) and a stationary phase (growth slows or stops).
If your dataset includes those regions, ln(N) versus t will curve rather than form a line, and a single μ estimated from one line
can be misleading. For best results, select only the time range where ln(N) versus t is approximately linear.
Two-point method (t1, N1) and (t2, N2)
With two measurements in log phase, μ is computed directly from the slope between the two points:
\[
\mu = \frac{\ln(N_2) - \ln(N_1)}{t_2 - t_1}
\]
This method is quick and common in lab calculations. It assumes the interval from t1 to t2 lies fully in log phase.
Multi-point best-fit method (linear regression on ln(N) vs t)
When you provide multiple measurements, the calculator fits a straight line:
ln(N) = a + μt,
where a is the intercept and μ is the slope. Using a best-fit line reduces the impact of noise in individual measurements.
\[
\mu = \frac{\sum (t_i - \bar{t})(y_i - \bar{y})}{\sum (t_i - \bar{t})^2},
\quad
a = \bar{y} - \mu \bar{t},
\quad
\text{where } y_i=\ln(N_i)
\]
The calculator also reports R², which measures how well the line explains variation in ln(N). Values close to 1 indicate a very
linear log-phase trend.
Generation time and doubling time
In the ideal exponential model, the generation time g equals the doubling time td.
They are related to μ by:
\[
g = t_d = \frac{\ln(2)}{\mu}
\]
If time is entered in hours, μ is reported in h-1 and g is reported in hours (with convenient conversions to minutes if needed).
Log basis option (ln vs log10)
The underlying biology model uses the natural log because N(t) = N0 · eμt.
However, many lab plots use log10. The calculator can display either ln(N) or log10(N) on the semi-log graph.
The slope changes by a constant factor:
\[
\log_{10}(N) = \frac{\ln(N)}{\ln(10)}
\]
So if the fitted slope on the log10 plot is s, then μ = s · ln(10). The calculator always computes μ from ln internally and then adjusts
the displayed slope when showing log10.
Interpreting the semi-log plot
On the semi-log plot, each point is (t, ln(N)) (or (t, log10(N)) for display). In log phase, the points should lie close to a line.
The slope badge on the plot reports μ (and the displayed slope if you choose log10), and hovering shows predicted values from the fitted line.
