π When Is a Log Model Appropriate?
A logarithmic model is appropriate when data shows a large initial rate of change that gradually decreases β rapid growth at first, then slowing down. Real-world examples: salaries, earthquakes (Richter scale), sound (decibels), acidity (pH).
Key signal in a table: If the input values appear to change proportionally (multiplicatively) over equal-length output value intervals, then a logarithmic model is appropriate. (This is the reverse of what we look for in an exponential model.)
y = a + b ln x or y = a + b log x
where a and b are constants and b β 0. Use ln for natural base models, log for base-10 models.
π Steps to Build a Log Model from Two Points
1
Substitute each (x, y) data point into the model to get two equations
2
Subtract one equation from the other to eliminate a
3
Solve for b using log properties (factor out b)
4
Back-substitute to find a
βοΈ Example 1 β Build L(x) = a + b ln x from Two Points
L(x) = a + b ln x with L(2) = 3 and L(5) = 7
| x | L(x) |
|---|---|
| 2 | 3 |
| 5 | 7 |
Step 1 β Write two equations:
L(2) = a + b ln 2 = 3
L(5) = a + b ln 5 = 7
Step 2 β Subtract to eliminate a:
(a + b ln 5) β (a + b ln 2) = 7 β 3
b(ln 5 β ln 2) = 4
Step 3 β Solve for b:
b = 4 / (ln 5 β ln 2) = 4 / ln(5/2) β 4.3654
Step 4 β Back-substitute for a:
a = 3 β b ln 2 β 3 β (4.3654)(0.6931) β β0.0258
L(2) = a + b ln 2 = 3
L(5) = a + b ln 5 = 7
Step 2 β Subtract to eliminate a:
(a + b ln 5) β (a + b ln 2) = 7 β 3
b(ln 5 β ln 2) = 4
Step 3 β Solve for b:
b = 4 / (ln 5 β ln 2) = 4 / ln(5/2) β 4.3654
Step 4 β Back-substitute for a:
a = 3 β b ln 2 β 3 β (4.3654)(0.6931) β β0.0258
βοΈ Example 2 β Pastor Salary Model (Regression)
S(p) = a + b ln p β Church Size vs Average Salary
| Church Size (p) | 75 | 120 | 250 | 300 | 450 | 650 | 900 | 1500 | 2500 | 5000 |
|---|---|---|---|---|---|---|---|---|---|---|
| Avg Salary ($1000s) | 42 | 58 | 70 | 79 | 88 | 95 | 110 | 125 | 138 | 175 |
(a) Regression on calculator (LnReg): a β β91.54 b β 29.91
S(p) = β91.54 + 29.91 ln p
(b) Predicted salary for church size 500:
S(500) = β91.54 + 29.91 ln(500) β $94,330 (β 94.3 thousand dollars)
S(p) = β91.54 + 29.91 ln p
(b) Predicted salary for church size 500:
S(500) = β91.54 + 29.91 ln(500) β $94,330 (β 94.3 thousand dollars)
Calculator steps: Enter data in L1 and L2 β STAT β CALC β 9:LnReg β Enter L1, L2, Yβ β Calculate
βοΈ Example 3 β Decibel Formula (Sound Intensity)
Ξ²(I) = a + b log(I) β Sound Intensity Model
Noisy traffic: I = 1Γ10β»β΅ W/mΒ² β 70 dB. Loud concert: I = 1 W/mΒ² β 120 dB.
Step 1 β Write two equations:
Ξ²(1Γ10β»β΅) = a + b log(10β»β΅) = a + b(β5) = 70
Ξ²(1) = a + b log(1) = a + b(0) = 120 β a = 120
Step 2 β Substitute a = 120 into first equation:
120 β 5b = 70 β β5b = β50 β b = 10
Model: Ξ²(I) = 120 + 10 log(I)
Step 3 β Predict: eardrum bursts at I = 1Γ10β΄ W/mΒ²:
Ξ²(10β΄) = 120 + 10 log(10β΄) = 120 + 10(4) = 160 dB
Ξ²(1Γ10β»β΅) = a + b log(10β»β΅) = a + b(β5) = 70
Ξ²(1) = a + b log(1) = a + b(0) = 120 β a = 120
Step 2 β Substitute a = 120 into first equation:
120 β 5b = 70 β β5b = β50 β b = 10
Model: Ξ²(I) = 120 + 10 log(I)
Step 3 β Predict: eardrum bursts at I = 1Γ10β΄ W/mΒ²:
Ξ²(10β΄) = 120 + 10 log(10β΄) = 120 + 10(4) = 160 dB
π΅ Fun Fact β The Word "Decibel"
deci- refers to the fact that the sound level is multiplied by 10 in the formula. -bel references Alexander Graham Bell, inventor of the telephone, for whom decibels are named.
Pattern recap: log(1) = 0 is a useful trick β when you substitute a data point where x = 1, the log term vanishes and you can immediately read off the value of a. Then substitute back to find b.
Card 1 of 6
Concept
When is a logarithmic model appropriate for a data set?
Tap to reveal β¨
Answer
When input values change proportionally over equal-length output intervals β rapid initial growth that gradually slows
The reverse of an exponential: x values multiply while y values add equally. Data curves steeply at first then flattens.
Topic 2.14
Logarithmic Function Context & Data Modeling
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Question 1 of 5
L(x) = a + b ln x, with L(2) = 3 and L(5) = 7. What equations can be used to find a and b?
Question 2 of 5
Using L(2) = a + b ln 2 = 3 and L(5) = a + b ln 5 = 7, which expression gives the value of b?
Subtract the two equations to eliminate a
Question 3 of 5
The sound intensity model is Ξ²(I) = a + b log(I). A loud concert at I = 1 W/mΒ² corresponds to 120 dB. What is the value of a?
log(1) = 0
Question 4 of 5
Using Ξ²(I) = 120 + 10 log(I), what is the predicted sound level in decibels for I = 1Γ10β΄ W/mΒ²?
log(1Γ10β΄) = log(10β΄) = 4
Question 5 of 5
A logarithmic regression gives S(p) = β91.54 + 29.91 ln p. What is the predicted average salary (in thousands) for a church size of 500?
Use a calculator: ln(500) β 6.2146
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