19.1: M1.01- Introduction

Last updated
Save as PDF

Page ID: 51684

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Learning Objectives

Be able to evaluate “power function” modeling formulas that give output values proportional to a constant power of the input value.
Be able to find the best-fit power and scale parameters implied by a dataset.
Be able to find the best-fit inverse function of data with a power-function relationship.
Be able to use common logarithms to compress the range of one or both variables in a relationship so that a more informative graph can be produced of data with a large dynamic range.
Be able to create semi-log and log-log graphs of data, to determine when such graphs are appropriate, and to read data that is presented in such graphs.
Be able to determine whether data variables have an exponential relationship by examining a semi-logarithmic graph of the data points.
Be able to determine whether data variables have a power-function relationship by examining a log-log graph of the data points.

Overview

Some situations exist in which all the input or output values are positive, but the ratio between the largest and smallest values (the dynamic range) is very large. This usually makes it impossible to for a regular graph to show the details of the shape of the relationship. Logarithms are a standard mathematical tool that has been developed to address this issue.

Graphing software usually supports modes in which one or both of the axes are graphed with logarithmic spacing rather than the usual uniform spacing. Such graphs can display a much wider range of values, and are often used in application areas that produce data with a large dynamic range.

Because of the way exponential and power functions make use of exponents in their formulas, appropriate logarithmic graphs of data with such relationships form straight lines, which are very easy to recognize and use for estimation. This also makes it easy to detect outliers in such relationships.

In addition to being a function that can be directly used in modeling, Logarithms have special connections to two other modeling functions: the exponential function we have already discussed and the power function that often arises as a result of dimensional relationships. This is be because the logarithm of an exponential function is a straight line, and a power function graphs as a straight line if the logarithms of both the x and y values are used.

Graphs are often provided with semi-logarithmic and logarithmic scales to take advantage of these relationships, which make it easy to recognize when data has an exponential or power-function relationship. Spreadsheet can produce such graphs automatically.

Mathematics for Modeling. Authored by: Mary Parker and Hunter Ellinger. License: CC BY: Attribution