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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

Discuss the advantages of using logarithmic scales for certain graphs
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 . 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.
It should be noted that the examples in the graphs were meant to illustrate a point and that the functions graphed were not necessarily unwieldy on a linearly scales set of axes.
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:
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Key Term Reference
 area
 Appears in these related concepts: Area Between Curves, Introduction to Circles, and Pedagogical Problem Solving Tasks
 constant
 Appears in these related concepts: Inverse Variation, Graphing Quadratic Equations in Vertex Form, and Direct Variation
 dependent variable
 Appears in these related concepts: The Cartesian System, What is a Quadratic Function?, and Converting between Exponential and Logarithmic Equations
 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, Linear Mathematical Models , and The Distance Formula and Midpoints of Segments
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 function
 Appears in these related concepts: Solving Differential Equations, Functions and Their Notation, and The Vertical Line Test
 graph
 Appears in these related concepts: Reading Points on a Graph, Graphing Equations, and Graphical Representations of Functions
 independent variable
 Appears in these related concepts: Experimental Design, Formulating the Hypothesis, and Experimental Research
 linear
 Appears in these related concepts: Factoring General Quadratics, Exponential Growth and Decay, and Graphs of Linear Inequalities
 point
 Appears in these related concepts: The Intermediate Value Theorem, Graphing Quadratic Equations In Standard Form, and Polynomial and Rational Functions as Models
 range
 Appears in these related concepts: Introduction to Domain and Range, The Derivative as a Function, and Visualizing Domain and Range
 set
 Appears in these related concepts: Sequences of Mathematical Statements, Sequences, and Introduction to Sequences
 variable
 Appears in these related concepts: What is a Linear Function?, Math Review, and Introduction to Variables
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Source: Boundless. “Solving Problems with Logarithmic Graphs.” Boundless Algebra. Boundless, 14 Oct. 2016. Retrieved 26 Oct. 2016 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/exponentslogarithmsandinversefunctions8/graphsofexponentialandlogarithmicfunctions354/solvingproblemswithlogarithmicgraphs1805878/