Logarithmic Models (College Board AP® Precalculus): Study Guide

Logarithmic models

When is a logarithmic model appropriate?

• Logarithmic functions are the inverses of exponential functions

  • so they model the 'reverse' type of relationship

• An exponential function has

  • outputs that change proportionally (multiplicatively)

    • as inputs change additively (by equal amounts)

• A logarithmic function has the opposite behavior

  • input values change proportionally

    • over equal-length output-value intervals

  • In other words, the inputs must be multiplied by a constant factor

    • to produce equal increases in the output

  • E.g. the Richter scale for earthquake magnitude is logarithmic

    • Each increase of 1 on the scale corresponds to a tenfold increase in the measured amplitude of the earthquake

How do I construct a logarithmic model from two data points?

• A logarithmic model can be constructed from two input-output pairs using the form $f\left(x\right)=a+b\ln (x+c)$

  • where $c$ is often determined by the context

    • commonly $c=0$ or $c=1$

  • On the exam, a question typically provides the form of the model

    • e.g. $G\left(t\right)=a+b\ln (t+1)$

    • and asks you to find the constants $a$ and $b$

• Here is the general process (using $f\left(x\right)=a+b\ln (x+c)$ with data points $(x_1,y_1)$ and $(x_2,y_2)$)

  • Assume that $c$ is known (it is usually given on the exam)

  • Substitute each data point into the function to get two equations

    • $y_1=a+b\ln (x_1+c)$

    • $y_2=a+b\ln (x_2+c)$

  • Solve the system for $a$ and $b$

  • Often one of the data points will cause the argument inside one of the logarithms to be equal to 1

    • This simplifies things greatly since $\mathbf{ln}\left(1\right)\mathbf{=}\mathbf{0}$

    • It gives $a=y_1$ directly

• Example: A study uses test scores to track academic content retention. At $t=0$ months, the average score is 75 points. At $t=3$ months, the average score is 70.84 points. The model to be used is $R\left(t\right)=a+b\ln (t+1)$.

  • $R\left(0\right)=75$ so

    • $75=a+b\ln \left(0+1\right)\Rightarrow 75=a+b\ln \left(1\right)\Rightarrow a=75$

      • this uses $\ln \left(1\right)=0$ in the final step

  • And $R\left(3\right)=70.84$, so

    • $70.84=a+b\ln (3+1)\Rightarrow 70.84=75+b\ln \left(4\right)$

      • using the value of $a$ from the previous step

  • Solve for $b$

    • $-4.16=b\ln \left(4\right)\Rightarrow b=-\frac{4.16}{\ln \left(4\right)}\approx -3.001$

  • So the model is

    • $R\left(t\right)=75-3.001\ln (t+1)$

Can a logarithmic model also be constructed from a proportion and a zero?

• Yes, a logarithmic model can be constructed from:

  • An appropriate proportion

    • i.e. the multiplicative factor for inputs over equal output intervals

  • A real zero

    • an input value where the output is zero

• This connects to the inverse relationship with exponential functions

  • If an exponential function has base $b$ and initial value 1

    • then its inverse logarithmic function has base $b$ and a zero at $x=1$

• E.g. for a model in the form $f\left(x\right)={\log }_b\left(kx\right)$

  • suppose input values are multiplied by 2 for each increase by 1 in the output

    • and the output is 0 when $x=4$

  • The proportion, which determines the base of the logarithm, is $b=2$

    • inputs double per unit output increase

  • And the zero is at $x=4$, so $kx$ must equal 1 when $x=4$

    • because ${\log }_b\left(1\right)=0$

  • Therefore the model is

    • $f\left(x\right)={\log }_2\left(\frac{x}{4}\right)$

How can transformations be used to build a logarithmic model?

• Logarithmic models can be constructed by applying transformations to the base function $f\left(x\right)={\log }_{b}x$

• The transformations allows the function to match the characteristics of the context or data:

  • A vertical shift ${\log }_b\left(x\right)+k$

    • adds $k$ to all the output values

      • the graph goes through $\left(1,k\right)$ instead of $\left(1,0\right)$

      • the location of the zero will move

  • A horizontal shift ${\log }_b\left(x+h\right)$

    • shifts the graph by $h$ units to the left

      • the zero will occur at $\left(1-h,0\right)$ instead of at $\left(1,0\right)$

  • A vertical dilation $a{\log }_{b}x$

    • adjusts the rate of growth

How do I construct a logarithmic model using technology?

• When data does not fit a simple logarithmic form exactly

  • a graphing calculator can fit a logarithmic regression model to the data

• This is done using the LnReg (or similar) function on most graphing calculators

  • The calculator determines the best-fit values for the constants $a$ and $b$

  • in the model $a+b\ln x$

Why is the natural logarithm often used in modeling?

• The natural logarithm ($\ln$, with base $e\approx 2.718$) appears frequently in real-world models

  • Many natural processes involve continuous growth or decay

    • and the natural base $e$ arises naturally in these contexts

• In logarithmic modeling questions on the exam, $\ln$ appears far more regularly than other logarithm bases

How can a logarithmic model be used to make predictions?

• Once a logarithmic model is constructed, it can be used to predict values of the dependent variable

  • by substituting input values into the function

• You can also predict input values

  • by substituting output values into the function

  • and solving to find the corresponding input value

• Some things to keep in mind:

  • Logarithmic functions grow very slowly for large inputs

    • Predictions for very large input values may not change much

  • The model may not remain valid beyond the range of data used to construct it

    • Just as with any model