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2.14 · When Is Log Model Appropriate?
What data pattern signals that a logarithmic model is appropriate?
Think about the rate of change over time
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2.14 · General Form
Write the general forms of a logarithmic function model.
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2.14 · Steps: Two-Point Model
What are the 4 steps to build a log model y = a + b ln x from two data points?
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2.14 · Apply: Write Equations
L(x) = a + b ln x, with L(2) = 3 and L(5) = 7. Write the two equations for a and b.
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2.14 · Apply: Solve for b
Given a + b ln 2 = 3 and a + b ln 5 = 7, what is the value of b?
Subtract to eliminate a
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2.14 · Apply: Solve for a
With b = 4/(ln5 − ln2) ≈ 4.3654, find a using L(2) = 3.
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2.14 · Useful Trick
If one data point is (1, y₀) and the model is y = a + b log(x), what does this tell you immediately?
What is log(1)?
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2.14 · Apply: Decibels
β(I) = a + b log(I). Traffic at I = 10⁻⁵ gives 70 dB. Concert at I = 1 gives 120 dB. Find a and b.
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✓ General Form
y = a + b ln x or y = a + b log x (b ≠ 0)
Use ln for natural base models, log (base 10) for models where decibels, pH, or Richter scale are involved.
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✓ When Log Model Is Appropriate
Input values change proportionally over equal-length output intervals — rapid initial growth that gradually decreases
Data curves steeply at first then flattens. Real-world: salaries, earthquakes, sound levels, pH.
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✓ Write Equations
a + b ln 2 = 3 and a + b ln 5 = 7
Substitute each (x, y) pair directly into L(x) = a + b ln x. Two data points give two equations in two unknowns.
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✓ Steps: Two-Point Model
1. Substitute each point 2. Subtract to eliminate a 3. Solve for b 4. Back-sub for a
Subtracting eliminates the constant a, leaving only b times a log expression = difference of y-values.
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✓ Solve for a
a = 3 − b ln 2 ≈ 3 − (4.3654)(0.6931) ≈ −0.0258
Substitute b and x=2, y=3 back into a + b ln 2 = 3. Solve: a = 3 − b ln 2.
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✓ Solve for b
b = 4 / (ln 5 − ln 2) = 4 / ln(5/2) ≈ 4.3654
Subtract equations: b(ln5 − ln2) = 4. Factor b out and divide both sides.
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✓ Decibels
a = 120 b = 10
I=1: a+b(0)=a=120. Then 120−5b=70 → b=10. Model: β(I) = 120 + 10 log(I).
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✓ Useful Trick: log(1) = 0
a = y₀ immediately
y = a + b·log(1) = a + b(0) = a. The b term vanishes, so the y-value at x=1 gives a directly.
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2.14 · Apply: Predict
β(I) = 120 + 10 log(I). Predict the sound level at I = 10⁴ W/m².
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2.14 · Regression
What calculator regression type do you use to find a log model y = a + b ln x?
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2.14 · Real-World Uses
Name four real-world phenomena that follow logarithmic patterns.
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2.14 · Predict from Regression
S(p) = −91.54 + 29.91 ln p models pastor salary in thousands. Predict S(500).
ln(500) ≈ 6.2146
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✓ Regression Type
LnReg (Logarithmic Regression)
STAT → CALC → 9:LnReg. Enter L1 (x), L2 (y), Y₁. This fits y = a + b ln x to the data.
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✓ Predict: 160 dB
β(10⁴) = 120 + 10 log(10⁴) = 120 + 40 = 160 dB
log(10⁴) = 4 since 10⁴ = 10,000. This is the threshold at which the human eardrum can burst.
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✓ Predict: S(500)
S(500) ≈ −91.54 + 29.91(6.2146) ≈ 94.3 thousand
≈ $94,300 predicted average pastor salary for a church of 500 people.
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✓ Real-World Uses
Sound (decibels), earthquakes (Richter scale), acidity (pH), salaries vs. organization size
All follow logarithmic patterns: rapid initial change that slows as the input grows large.