Notes on Semi-Log Plots in Mathematics

LESSON 2.15: SEMI-LOG PLOTS

LESSON OBJECTIVE

  • 2.B: Construct equivalent graphical, numerical, analytical, and verbal representations of functions that are useful in a given mathematical or applied context, with and without technology.

  • 3.C: Support conclusions or choices with a logical rationale or appropriate data.

INTRODUCTION TO SEMI-LOG PLOTS

  • Previous topic (Topic 2.9 - Logarithmic Expressions): discussion about logarithmic scaled axes for understanding exponential data.

  • Current focus: semi-log plots, which provide a more detailed examination of logarithmically scaled data.

DEFINITION OF SEMI-LOG PLOTS

  • A semi-log plot is defined as a graphical representation where one of the axes (typically the y-axis) is logarithmically scaled.

  • In AP Precalculus, the vertical (y) axis is the axis that is logarithmically scaled.

  • Purpose: To visualize exponential functions in a linear format, making them easier to interpret.

FEATURES OF SEMI-LOG PLOTS

  • In a semi-log plot with a logarithmically scaled y-axis, exponential functions will appear linear.

  • Examples include population growth:
      - Graphs displayed: population of English Americans in the United States from 1620 to 1820.
      - For t = 0, the year is 1620.
      - Left graph: normal scale on the vertical axis.
      - Right graph: semi-log plot demonstrating logarithmic scaling on the vertical axis.

EXAMPLE ANALYSIS

EX 1: Justifying Exponential Models

  • Question: Justify why an exponential model is appropriate for the population of English Americans in the U.S. from 1620-1820 using the semi-log plot.

  • Answer: The semi-log plot shows that data from 1660 onwards appears linear, which indicates that the function governing population growth is exponential. Therefore, an exponential model is appropriate.

EX 2: Modeling Joke Spread

  • Scenario: After a math joke, the spread of how many people, P, that have heard it after t minutes is graphed on a semi-log plot.

  • Options for functions that could model P:
      - (A) P(t) = 2 + 2t (not valid)
      - (B) P(t) = 2 + 2t (not valid)
      - (C) P(t) = 2 + log₂t (not valid)
      - (D) P(t) = 2(2) (correct)

  • The semi-log plot indicates that an exponential equation can model the growth, as shown by the linear appearance of the data.

IMPORTANT NOTE ABOUT SEMI-LOG PLOTS

  • Clarification: Logarithmically scaling the vertical axis does not alter the actual y-values of the data.

  • It means that equally spaced values on the logarithmic y-axis are proportional rather than linear.

EXAMPLE ANALYSIS CONTINUED

EX 3: Graphing Selected Values

  • Data Given:
      - If a table lists values where x takes on the values 1, 2, 3, 4, 5, and f(x) has values 40, 60, 90, 135, 203, identify which graph corresponds to the data on a semi-log plot.

  • Choices:
      - (A)
      - (B) Linear Scale
      - (C) Linear Scale
      - (D) Logical options to consider.

EX 4: Plotting Points

  • Instructions: Plot specified points A(0, 200), B(1, 25), C(2, 7), D(3.2, 800), E(4.6, 1.5) on the same coordinate plane to visually examine exponential growth behavior on a semi-log plot.

LINEAR MODELS FOR SEMI-LOG PLOTS

  • General Formulation:
      - Given the exponential model y=abxy = ab^x, the corresponding linear model for the semi-log plot is expressed as
      y=(extlognb)x+extlognay = ( ext{log}_n b)x + ext{log}_n a, where n>0n > 0 and n+1n + 1 (n is the proportional counting value).

  • Important Note:
      - The slope of our linear function corresponds to extlogbext{log} b and the y-intercept corresponds to extlogaext{log} a.

EX 5: Modeling the Data

Part (a): Linear Model Equation

  • Create an equation for the linear model of the semi-log plot from a given data table:
      1. Formulation: y=(extlogb)x+extlogay = ( ext{log} b)x + ext{log} a.
      2. Example: Suppose AROC = extlog60−extlog40ext{log} 60 - ext{log} 40.
      3. Calculation:
        - extAROC=0.17609…ext{AROC} = 0.17609….

Part (b): Exponential Model Equation

  • Use the linear model from Part (a) to write the equation of the exponential model:
      - Given values computed from log transformation:
        - extloga=1.4259ext{log} a = 1.4259, \
        - extlogb=0.17609ext{log} b = 0.17609,
      - From these, establish values for A and b:
        - A=101.4259A = 10^{1.4259}
        - b=100.17609b = 10^{0.17609}
        - Resulting exponential model: y=20(1.5)xy = 20(1.5)^x.

CONCLUSION

  • Understanding the relationship between linear and semi-log plots allows for effective modeling of exponential data in applied mathematics. This knowledge is essential not only academically but also in practical scenarios involving growth phenomena such as population increases or spreading processes.


INTRODUCTION TO SEMI-LOG PLOTS

  • Previous topic involved logarithmic scales to help us understand data that grows or shrinks quickly, like population growth.

  • Now, we focus on semi-log plots, which are handy tools to visualize data that changes exponentially.

DEFINITION OF SEMI-LOG PLOTS

  • A semi-log plot is a type of graph where one axis (usually the vertical one) uses a logarithmic scale.

  • In simpler terms, this means that instead of evenly spaced numbers, the y-axis uses values that are multiplied or divided, which helps show large changes more clearly.

WHY USE SEMI-LOG PLOTS?

  • They help us see patterns in data that is growing rapidly. For example, if a population doubles every year, it will show as a straight line on a semi-log plot, making it easier to understand the growth trend.

HOW THEY LOOK

  • Imagine you plot the population of a town over years. The semi-log plot helps you see if and how fast the population is growing by turning curves into straight lines.

EXAMPLES

  • Population Growth: If you plot the population of English Americans in the U.S. from 1620 to 1820, the data appears linear on a semi-log plot after a certain year, showing that the growth was exponential.

  • Funny Joke Spread: When analyzing how fast people share a joke, the semi-log plot can help predict how quickly it spreads, again looking for a straight line indication of rapid growth.

KEY POINTS TO REMEMBER

  • Using a logarithmic scale changes how the numbers look but doesn’t change the actual data. Equal distances on the scale represent proportional changes, not just the same numerical difference.

  • The relationship between linear and semi-log plots is essential to effectively modeling exponential data and understanding things like population increase or how quickly news travels.