Semi-log Plots — AP® Precalculus Notes

Exponential data are hard to read on an ordinary graph: the early values are squashed near the axis and the later ones shoot off the top. A semi-log plot fixes that by putting the vertical axis on a logarithmic scale. This page shows why exponential data then become a straight line, how to read the model from the line, and what other data look like.

One logarithmic axis

On a semi-log plot the horizontal axis is ordinary and the vertical axis is a logarithmic scale with some base $n > 1$ , usually $10$ . The labels $1$ , $10$ , $100$ , $1{,}000$ are equally spaced, so equal vertical distances mean equal factors, not equal differences. Plotting $y$ on that axis is the same as plotting $\log y$ on an ordinary one.

Why exponential data make a line

Take $y = a \cdot b^x$ and apply the properties of logarithms:

$\log y = \log a + \log(b^x) = \log a + x\log b.$

That is a linear function of $x$ , with slope $\log b$ and intercept $\log a$ . So exponential data lie on a straight line on a semi-log plot. No vertical shift is involved: the form must be exactly $a \cdot b^x$ .

Data	Ordinary graph	Semi-log plot
Exponential $a \cdot b^x$ , $b > 1$	concave up curve	straight line, slope $\log b$
Linear $mx + k$ , $m > 0$	straight line	concave down curve
Shifted $a \cdot b^x + k$	concave up curve	curve, not a line

Reading the model from the line

Worked example 1. App users lie on a line through $(0, 20)$ and $(8, 2{,}000)$ on a base- $10$ semi-log plot. Find $y = a \cdot b^t$ .

Slope in powers of $10$ : $\frac{\log 2{,}000 - \log 20}{8 - 0}$ $= \frac{3.301 - 1.301}{8}$ $= 0.25$ .

So $\log b = 0.25$ and $b = 10^{0.25} \approx 1.778$ : about $77.8\%$ growth per year.

The intercept is $\log a = \log 20$ , so $a = 20$ . The model is $y = 20 \cdot 10^{t/4}$ .

Worked example 2. A table gives $\log y$ $= 0.5, 0.8, 1.1, 1.4, 1.7$ at $t = 0, 1, 2, 3, 4$ . Find $y$ .

$\log y$ rises $0.3$ per step, so $\log y = 0.5 + 0.3t$ .

Undo the logarithm: $y$ $= 10^{0.5 + 0.3t}$ $= 10^{0.5} \cdot (10^{0.3})^t$ $\approx 3.16 \cdot 2.00^t$ .

With a natural-log scale the same steps use $e$ : the line $\ln y = 1.2 + 0.05t$ gives $y = e^{1.2} \cdot e^{0.05t}$ .

Reading values between labels

Halfway up between $200$ and $2{,}000$ on the logarithmic axis is not $1{,}100$ . It is the value whose logarithm is halfway between $\log 200$ and $\log 2{,}000$ , about $2.801$ : the value is $\sqrt{200 \cdot 2{,}000}$ $= \sqrt{400{,}000}$ $\approx 632$ . On a log scale, "halfway" means the same factor up from the lower value as down from the upper one.

Comparing models on one plot

Several exponential models drawn on one semi-log plot are all straight lines. The steeper line has the larger growth factor, since slope $= \log b$ . The higher intercept has the larger starting value. Parallel lines grow by the same factor each step.