Solving Problems with Logarithmic Graphs
Some functions with rapidly changing shape are best plotted on a scale that increases exponentially, such as a logarithmic graph.
Learning Objective

Convert problems to logarithmic scales and discuss the advantages of doing so
Key Points
 Logarithmic graphs use logarithmic scales, in which the values differ exponentially. For example, instead of including marks at
$0,1,2$ and$3$ , a logarithmic scale may include marks at$0.1, 1, 10$ and$100$ , each an equal distance from the previous and next.  Logarithmic graphs allow one to plot a very large range of data without losing the shape of the graph.
 Logarithmic graphs make it easier to interpolate in areas that may be difficult to read on linear axes. For example, if the plot
$y=x^5$ is scaled to show a very wide range of$y$ values, the curvature near the origin may be indistinguishable on linear axes. It is much clearer on logarithmic axes.
Terms

logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number.

interpolate
To estimate the value of a function between two points between which it is tabulated.
Full Text
Why Use a Logarithmic Scale?
Many mathematical and physical relationships are functionally dependent on highorder variables. This means that for small changes in the independent variable there are very large changes in the dependent variable. Thus, it becomes difficult to graph such functions on the standard axis.
Consider, as an example, the StefanBoltzmann law, which relates the power (j^{*}) emitted by a black body to temperature (T).
On a standard graph, this equation can be quite unwieldy. The fourthdegree dependence on temperature means that power increases extremely quickly. The fact that the rate is everincreasing (and steeply so) means that changing scale (scaling the axes by
For very steep functions, it is possible to plot points more smoothly while retaining the integrity of the data: one can use a graph with a logarithmic scale, where instead of each space on a graph representing a constant increase, it represents an exponential increase. Where a normal (linear) graph might have equal intervals going 1, 2, 3, 4, a logarithmic scale would have those same equal intervals represent 1, 10, 100, 1000. Here are some examples of functions graphed on a linear scale, semilog and logarithmic scales.
The top left is a linear scale. The bottom right is a logarithmic scale. The top right and bottom left are called semilog scales because one axis is scaled linearly while the other is scaled using logarithms.
Logarithmic scale
The graphs of functions
As you can see, when both axis used a logarithmic scale (bottom right) the graph retained the properties of the original graph (top left) where both axis were scaled using a linear scale. That means that if we want to graph a function that is unwieldy on a linear scale we can use a logarithmic scale on each axis and retain the properties of the graph while at the same time making it easier to graph.
With the semilog scales, the functions have shapes that are skewed relative to the original. When only the
Converting Linear to Logarithmic Scales
The primary difference between the logarithmic and linear scales is that, while the difference in value between linear points of equal distance remains constant (that is, if the space from
Thus, if one wanted to convert a linear scale (with values
The advantages of using a logarithmic scale are twofold. Firstly, doing so allows one to plot a very large range of data without losing the shape of the graph. Secondly, it allows one to interpolate at any point on the plot, regardless of the range of the graph. Similar data plotted on a linear scale is less clear.
Solving Problems Using Logarithmic Graphs
A key point about using logarithmic graphs to solve problems is that they expand scales to the point at which large ranges of data make more sense. In the equation mentioned above (
Taking the logarithm of each side of the equations yields:
Recall the following properties of logarithms:
Using the above, our equation becomes:
Key Term Reference
 area
 Appears in these related concepts: Area Between Curves, Pedagogical Problem Solving Tasks, and Introduction to Circles
 constant
 Appears in these related concepts: Graphing Quadratic Equations in Vertex Form, Inverse Variation, and Direct Variation
 dependent variable
 Appears in these related concepts: Converting between Exponential and Logarithmic Equations, The Cartesian System, and What is a Quadratic Function?
 difference
 Appears in these related concepts: The Order of Operations, Factoring a Difference of Squares, and Basic Operations
 distance
 Appears in these related concepts: Inequalities with Absolute Value, The Distance Formula and Midpoints of Segments, and Linear Mathematical Models
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 exponential
 Appears in these related concepts: Exponential Growth and Decay, Solving General Problems with Logarithms and Exponents, and Population Growth
 function
 Appears in these related concepts: Functions and Their Notation, The Vertical Line Test, and What is a Linear Function?
 graph
 Appears in these related concepts: Graphical Representations of Functions, Graphing Equations, and Reading Points on a Graph
 independent variable
 Appears in these related concepts: Experimental Design, Experimental Research, and Formulating the Hypothesis
 interval
 Appears in these related concepts: Schedules of Reinforcement, The Intermediate Value Theorem, and Interval Notation
 linear
 Appears in these related concepts: SecondOrder Linear Equations, Graphs of Linear Inequalities, and Factoring General Quadratics
 point
 Appears in these related concepts: Relative Minima and Maxima, Polynomial and Rational Functions as Models, and Graphing Quadratic Equations In Standard Form
 range
 Appears in these related concepts: The Derivative as a Function, Visualizing Domain and Range, and Introduction to Domain and Range
 set
 Appears in these related concepts: Sequences, Introduction to Sequences, and Sequences of Mathematical Statements
 variable
 Appears in these related concepts: Introduction to Variables, Fundamentals of Statistics, and Math Review
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