AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/2/2-15)
Unit 2 · Topic 2.15
2.15 Semi-log Plots
A semi-log plot uses a logarithmic scale on the y-axis. On such a plot, exponential data line up in a straight line. That makes exponential patterns easy to spot, and you can fit a line to the logs and convert it back to an exponential model.
Key terms
- semi-log plot
- logarithmic scale
- linearization
- linear model
What a semi-log plot is
In a semi-log plot, one axis (here, the y-axis) is logarithmically scaled with some base n > 1, and the other is ordinary. On the y-axis, equal distances stand for equal factors, like 1, 10, 100, 1000 for base 10.
Plotting y on a log scale is the same as plotting log_n(y) on a regular scale.
Reading the axis
On a base-10 semi-log plot, the labeled marks 1, 10, 100, 1,000 are equally spaced. Between 10 and 100, the smaller marks for 20, 30, …, 90 crowd closer together as they go up. The mark for 20 sits about 30% of the way up (log 20 ≈ 1.301), while 90 sits about 95% of the way (log 90 ≈ 1.954).
If the plotted points rise along a straight line, the data grow exponentially. If the points bend downward and flatten, the data grow more slowly than exponential. If they bend upward, faster.
A semi-log plot can't show an output of 0 or a negative output, because those values have no logarithm.
Why exponential data look straight
Take the log of y = a · bˣ: log_n(y) = log_n(a) + x · log_n(b).
That's a linear function of x. Its slope is log_n(b) and its vertical intercept is log_n(a). So exponential data plot as a straight line on a semi-log plot.
This gives you a test: if the logs of the outputs trend linearly, an exponential model is appropriate. Even data that are exponential plus a constant often look linear on a semi-log plot for large inputs, so you don't need to find and subtract the constant first.
Going back and forth
From exponential to linear: y = a · bˣ becomes log_n(y) = (log_n b)x + log_n a.
From a semi-log line back to exponential: if log_n(y) = mx + c, then y = nᶜ · (nᵐ)ˣ. So a = nᶜ and b = nᵐ.
Use the same base on both trips. If the plot uses ln, exponentiate with e; if it uses log base 10, use 10.
Linear tools on log data
Everything you know about lines works on the semi-log plot: find the slope from two points, write point-slope form, or run a linear regression on (x, log y). Then convert back.
A positive slope on the semi-log plot means exponential growth (b > 1); a negative slope means decay (0 < b < 1).
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Exponential to semi-log
Write the linear equation that y = 4(1.5)ˣ becomes on a semi-log plot with base 10.
Show the solutionHide the solution
- Step 1: Take log base 10 of both sides: log y = log 4 + x log 1.5.
- Step 2: log 4 ≈ 0.602 and log 1.5 ≈ 0.176.
Answer: log y ≈ 0.602 + 0.176x (slope log 1.5, intercept log 4).
- Example 2Calculator allowed
Semi-log line to exponential model
On a semi-log plot with base 10, data fall along a line through (0, 0.7) and (5, 2.2), where the vertical coordinate is log y. Find an exponential model and predict y at x = 3.
Show the solutionHide the solution
- Step 1: Slope: (2.2 − 0.7)/(5 − 0) = 0.3. Intercept: 0.7. So log y = 0.3x + 0.7.
- Step 2: Convert: y = 10^0.7 · (10^0.3)ˣ.
- Step 3: 10^0.7 ≈ 5.012 and 10^0.3 ≈ 1.995.
- Step 4: At x = 3: log y = 0.3(3) + 0.7 = 1.6, so y = 10^1.6 ≈ 39.811.
Answer: y ≈ 5.012(1.995)ˣ; y(3) ≈ 39.811.
- Example 3Calculator allowed
Trap: matching the base
A semi-log plot uses ln y on its vertical axis, and the data lie on the line ln y = 1.2 + 0.05x. Write the exponential model.
Show the solutionHide the solution
- Step 1: The axis is natural log, so undo it with e, not 10.
- Step 2: y = e^1.2 · (e^0.05)ˣ.
- Step 3: e^1.2 ≈ 3.320 and e^0.05 ≈ 1.051.
- Step 4: Using 10 instead would give 10^1.2 ≈ 15.849 and 10^0.05 ≈ 1.122, a completely different model.
Answer: y ≈ 3.320(1.051)ˣ.
Common mistakes
- Converting back with the wrong base. Match the base of the log used on the axis.
- Using the slope of the semi-log line as the growth factor. The factor is n raised to the slope, not the slope itself.
- Reading y-values from a log axis as if it were evenly spaced.
On the exam
- Expect questions that show data on a semi-log plot and ask which model fits, or that give the semi-log line and ask for the exponential model's growth factor.
- If asked why a semi-log plot supports an exponential model, say the logarithms of the outputs are approximately linear in the input.
Connected topics
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Check yourself
4 questions on 2.15 Semi-log Plots. Pick an answer to see if you got it, and why.
A scientist plots data on a semi-log plot, with log₁₀ y on the vertical axis and x on the horizontal axis. The points lie close to the line log₁₀ y = 0.3x + 1.2. Which of the following exponential models is consistent with this line?
Four data sets are each graphed on a semi-log plot, with the vertical axis on a logarithmic scale. Which data set will appear closest to linear?
The function y = 4(1.5)ˣ is graphed on a semi-log plot, with log₁₀ y on the vertical axis and x on the horizontal axis. Which of the following is closest to the slope and vertical intercept of the resulting line?
A data set is graphed on a semi-log plot, with the vertical axis on a logarithmic scale. The points lie close to a line with a negative slope. Which of the following could model the data?
0 of 4 answered