Unit 4 · Topic 2.15

Semi-log Plots

Recognize that exponential data become approximately linear when the output axis is logarithmic. Develop the idea through symbolic, numerical, graphical, and contextual representations.

Learning Goals

Recognize exponential data as linear on a log-scaled output axis.
Linearize exponential models.
Recover exponential parameters from semi-log slope and intercept.

1. Essential Structure

Recognize that exponential data become approximately linear when the output axis is logarithmic.

\[y=ab^x\Rightarrow\log y=\log a+x\log b\]

Read the formula together with its domain, units, starting input, and the interval length over which change is measured.

2. Core Ideas

If \(y=ab^x\), then \(\log_ny=\log_na+x\log_nb\).
Semi-log slope \(m=\log_nb\) and intercept \(c=\log_na\), so \(b=n^m\) and \(a=n^c\).
Equal vertical distances on a log axis represent equal ratios.
Curvature on a semi-log plot weakens the case for an unshifted exponential model.

3. Graph and Representation

Equal horizontal steps produce equal vertical log changes, so exponential data align along a line.

Use graphical evidence together with an algebraic or numerical reason. A graphing window alone does not establish domain, end behavior, or model validity.

4. Original Worked Example

What pattern does \(y=5(2)^x\) make on a semi-log plot with logarithmic y-axis?

It forms a line with intercept \(\log5\) and slope \(\log2\).

Check the result in the original representation and explain its meaning, including units when the quantities are contextual.

5. AP Reasoning Workflow

Confirm positive outputs.
Transform outputs or use a log-scaled axis.
Fit a line.
Convert slope and intercept back to \(a\) and \(b\).
Validate in the original scale.

A strong AP response shows the mathematical evidence first and then states a precise conclusion.

6. Extended Example and Application

If \(\log_{10}y=0.301x+1.2\), then \(y=10^{1.2}(10^{0.301})^x\approx15.85(2.00)^x\), so outputs approximately double each unit.

7. Technology and Validation

Use linear regression on transformed outputs and report the final model in the original y-scale.

When technology is used, record the model or result, preserve sufficient precision, and explain why the output answers the question.

8. Understand What a Semi-Log Plot Displays

A semi-log plot uses a linear scale on one axis and a logarithmic scale on the other. In this course, the input axis is linear and the positive output axis uses logarithmic base \(n>1\).

Axis	Equal visual spacing represents
Linear x-axis	Equal additive changes in \(x\)
Logarithmic y-axis	Equal multiplicative changes in \(y\)

On a base-10 output axis, \(1,10,100,1000\) appear at equal vertical intervals because each value is 10 times the preceding value. Only positive outputs can appear because a real logarithm is undefined for zero and negative values.

AP scope: use a logarithmically scaled y-axis to linearize exponential behavior. Using a logarithmic x-axis to linearize a logarithmic function is outside this topic's required scope.

9. Explain Why Exponential Data Become Linear

Start with an exponential model

\[y=ab^x,\qquad a>0,\ b>0,\ b\ne1.\]

Taking a base-\(n\) logarithm gives

\[\log_ny=\log_na+x\log_nb.\]

If \(z=\log_ny\), then

\[z=mx+c,\qquad m=\log_nb,\quad c=\log_na.\]

An exponential relationship becomes linear between \(x\) and \(\log_ny\). Growth with \(b>1\) produces a positive slope; decay with \(0<b<1\) produces a negative slope. Changing the log base changes the numerical slope and intercept, but not whether the transformed points are collinear.

10. Construct a Linearized Data Table

Consider \(y=5(2)^x\). Add a transformed-output row before fitting a line.

\(x\)	0	1	2	3	4
\(y\)	5	10	20	40	80
\(\log_{10}y\)	0.6990	1.0000	1.3010	1.6021	1.9031

The transformed outputs increase by approximately \(0.3010\) whenever \(x\) increases by 1. Their exact linear equation is

\[\log_{10}y=(\log_{10}2)x+\log_{10}5.\]

Plotting these transformed points on ordinary linear axes and plotting the original points on a base-10 semi-log graph display the same linear structure.

11. Recover an Exponential Model from the Semi-Log Line

Suppose the transformed regression line is

\[z=mx+c,\qquad z=\log_ny.\]

Exponentiate to return to the original scale:

\[y=n^{mx+c}=n^c(n^m)^x.\]

Thus \(a=n^c\) and \(b=n^m\). For example, if

\[\log_{10}y=0.176091x+1.301030,\]

then \(a=10^{1.301030}\approx20\), \(b=10^{0.176091}\approx1.5\), and

\[y\approx20(1.5)^x.\]

Important: the semi-log slope is \(\log_nb\), not the growth factor \(b\). Exponentiate the slope to recover the factor.

12. Read Values between Powers on a Logarithmic Axis

Values between labeled powers are positioned according to their logarithms, not their ordinary numerical distance. Within the base-10 interval from 10 to 100:

Original value	Log coordinate	Position through the interval
10	1	0%
20	1.3010	about 30.1%
50	1.6990	about 69.9%
100	2	100%

The vertical midpoint is the geometric mean, not the arithmetic mean:

\[10^{1.5}=\sqrt{10\cdot100}\approx31.62.\]

Equal distances on the axis represent equal ratios. Moving halfway through one decade multiplies the starting value by \(\sqrt{10}\).

13. Detect an Additively Shifted Exponential Pattern

An additively shifted exponential, \(y=ab^x+k\), is generally curved on a semi-log plot because a logarithm does not distribute over addition. Consider

\[y=3(2)^x+5.\]

\(x\)	0	1	2	3	4	5
\(y\)	8	11	17	29	53	101
\(\log_{10}y\)	0.9031	1.0414	1.2304	1.4624	1.7243	2.0043

The transformed differences approach \(\log_{10}2\approx0.3010\) as \(x\) grows. The constant 5 becomes small compared with \(3(2)^x\), so the transformed points trend toward a line. This large-input behavior can support an additively transformed exponential model without first guessing and subtracting the shift.

If the shift is known, the exact linearization is

\[\log_{10}(y-5)=\log_{10}3+x\log_{10}2.\]

14. Complete a Semi-Log Modeling Workflow

Suppose positive data produce the base-10 transformed regression

\[\log_{10}y=0.176091x+1.301030.\]

Identify: the transformed points are approximately linear with no systematic residual pattern.
Recover parameters: \(a=10^{1.301030}\approx20\) and \(b=10^{0.176091}\approx1.5\).
State the original model: \(\hat y=20(1.5)^x\).
Predict: at \(x=6\), \(\hat y=20(1.5)^6\approx227.81\).
Validate: calculate residuals \(y-\hat y\) in the original units and check for structure, unusual points, and changing spread.

Linear association after transformation is evidence for an exponential model, not proof that the context remains exponential forever. Check the original scatterplot, transformed plot, residuals, domain, and reasonableness before interpolating or extrapolating.

AP-ready conclusion: name the transformed variables, explain the slope and intercept, return the equation to the original scale, and interpret the final result with units and context.

15. Common Errors

Do not interpret the plotted vertical coordinate as the original y-value.
Reading equal log-axis spacing as equal original-unit differences.
Taking real logarithms of zero or negative outputs.
Giving a numerical result without a domain check, units, or interpretation.

Key Takeaways

Recognize that exponential data become approximately linear when the output axis is logarithmic.
Core relationship: \(y=ab^x\Rightarrow\log y=\log a+x\log b\)
Error check: Do not interpret the plotted vertical coordinate as the original y-value.

Checkpoint · Topic 2.15

Linearize \(y=12(1.5)^x\) with natural logs.
Recover \(a\) and \(b\) from \(\log_2y=3+0.5x\).
Explain what a curved semi-log plot indicates.