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2.15 · Definition
What is a semi-log plot? Which axis is log-scaled in AP Precalculus?
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2.15 · Key Fact
What happens to an exponential function when plotted on a semi-log plot (log y-axis)?
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2.15 · Justify Model
How do you use a semi-log plot to justify an exponential model for data?
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2.15 · Important Note
When the y-axis is logarithmically scaled, do the y-values of data points change?
Think carefully!
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2.15 · Log Scale Spacing
On a log-scaled y-axis with base 10, what do equally-spaced gridlines represent?
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2.15 · Linear Model
For y = ab^x, what is the slope and y-intercept of the corresponding linear model on a semi-log plot?
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2.15 · Apply: Find Linear Model
Data: f(1)=40, f(2)=60. Find the slope of the linear semi-log model (base 10).
slope = (log 60 − log 40) / (2 − 1)
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2.15 · Apply: Recover a and b
Linear semi-log model (base 10): y = 0.17609x + 1.4259. Find a and b for y = ab^x.
a = 10^(y-intercept) b = 10^(slope)
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✓ Key Fact
It appears linear — a straight line on the semi-log plot
Taking log of y = ab^x gives log y = log a + x·log b, which is linear in x. Exponential → linear on semi-log.
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✓ Definition
A plot where the vertical (y) axis is logarithmically scaled; x-axis stays linear
Equally-spaced gridlines on the y-axis correspond to proportional values (1, 10, 100, 1000). The actual y-values of data do NOT change.
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✓ Important Note
NO — the y-values do NOT change
Scaling only changes the visual spacing of gridlines. A data point with y=50 is still plotted at y=50, just positioned between the 10 and 100 gridlines.
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✓ Justify Model
If the semi-log plot appears linear, the data is exponential
Example: English American population data appears linear on a semi-log plot after t=40 → exponential model is appropriate.
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✓ Linear Model
Slope = logn(b) y-intercept = logn(a)
Where n = base of the log scale. To recover: a = n^(y-intercept), b = n^(slope). The linear model is y = logn(b)·x + logn(a).
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✓ Log Scale Spacing
Powers of 10: 1, 10, 100, 1000 …
Each equally-spaced gridline multiplies by 10. This is why exponential curves become straight lines — each unit of x multiplies y by the same factor.
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✓ Recover a and b
b = 10^0.17609 ≈ 1.5 a = 10^1.4259 ≈ 26.67
Exponential model: y ≈ 26.67(1.5)^x, or equivalently y = (80/3)(1.5)^x.
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✓ Slope of Linear Model
slope ≈ 0.17609
(log 60 − log 40) / (2−1) = log(60/40) = log(1.5) ≈ 0.17609. This equals log10(b), so b = 10^0.17609 ≈ 1.5.
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2.15 · Identify Function from Semi-log
A semi-log plot is linear. The line equation is log P = b + mt. Which function form models P?
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2.15 · Reading Semi-log
A point with y = 50 is plotted on a base-10 semi-log graph. Between which gridlines does it appear?
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2.15 · Correct Graph
Which graph correctly shows a semi-log plot of exponential data: one with y-axis labels 0,1,2,3 or one with y-axis labels 1,10,100,1000?
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2.15 · Why Semi-log?
Why is a semi-log plot useful for displaying exponential data?
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✓ Reading Semi-log
Between the 10 and 100 gridlines (since 10 < 50 < 100)
log(50) ≈ 1.699, which is between 1 (=log10) and 2 (=log100). The point plots 69.9% of the way from the 10-line to the 100-line.
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✓ Function Form
P(t) = 10^(b+mt) = 10^b · (10^m)^t — exponential
This matches P = a·r^t where a = 10^b and r = 10^m. The linear semi-log model always corresponds to an exponential function.
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✓ Why Semi-log?
Exponential data becomes linear, making it easier to see patterns and fit a model
On a normal scale, very large values dominate and small values are invisible. The semi-log plot spaces all values proportionally so all data points are distinguishable.
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✓ Correct Graph
The one with y-axis labels 1, 10, 100, 1000 (log scale)
A semi-log plot requires the y-axis to be logarithmically scaled — gridlines at powers of the base. Labels like 0,1,2,3 are linear, not log-scaled.