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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 scales make up logarithmic graphs.
Learning Objective

Discover 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 of 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
Many mathematical and physical relationships are functionally dependent on highorder variables.
Consider 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 is of little help in making the graph easier to interpret.
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 .
Logarithmic Scale
The graphs of functions f(x)=10^x, f(x)=x, and f(x)=log(x) on four different coordinate plots. Note how each function changes shape on each set of coordinates.
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 0 to 1 on the scale is 1cm on the page, the distance from 1 to 2, 2 to 3, etc., will be the same), the difference in value between points on a logarithmic scale will change exponentially. A logarithmic scale will start at a certain power of 10, and with every unit will increase by a power of 10.
Thus, if one wanted to convert a linear scale (with values 05) to a logarithmic scale, one option would be to replace 0, 1, 2, 3, 4, and 5 with 0.001, 0.01, 0.1, 1, 10, and 100, respectively. Between each major value on the logarithmic scale, the hashmarks become increasingly closer together with increasing value. For example, in the space between 1 and 10, the 8 and 9 are much closer together than the 2 and 3.
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 .
Points of log(y) on a Linear Scale
Notice how values of y less than 10 are indistinguibile.
Graph of log(y) on a SemiLog Scale
Both plots capture y well for their respective ranges, but note how easily distinguibile the points are in both the lower and higher areas.
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 logarithms, the product of numbers is the sum of their logarithms. In the equation mentioned above, plotting j vs. T would generate the expected curve, but the scale would be such that minute changes go unnoticed and the large scale effects of the relationship dominate the graph—it is so big that the "interesting areas" won't fit on the paper on a readable scale.
Taking logarithms, however, one ends with
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Key Term Reference
 constant
 Appears in these related concepts: Inverse Variation, Combined Variation, and Direct Variation
 distance
 Appears in these related concepts: Inequalities with Absolute Value, Symmetry, and The Distance Formula and Midpoints of Segments
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 function
 Appears in these related concepts: Inverse Functions, Solving Differential Equations, and Functions and Their Notation
 graph
 Appears in these related concepts: Graphing on Computers and Calculators, Reading Points on a Graph, and Graphing Functions
 linear
 Appears in these related concepts: Trinomials of the Form ax^2 + bx + c, Where a is Not Equal to 1, Exponential Growth and Decay, and Graphs of Linear Inequalities
 point
 Appears in these related concepts: The Intermediate Value Theorem, Quadratic Functions of the Form f(x) = ax^2 + bx + c, Where a is not Equal to 0, and Graphing Equations
 range
 Appears in these related concepts: Range, The Derivative as a Function, and Visualizing Domain and Range
 slope
 Appears in these related concepts: Making Inferences About the Slope, Slope and Intercept, and Applications of Linear Functions and Slope
 variable
 Appears in these related concepts: Related Rates, Math Review, and Psychology and the Scientific Method: From Theory to Conclusion
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Cite This Source
Source: Boundless. “Solving Problems with Logarithmic Graphs.” Boundless Algebra. Boundless, 21 Jul. 2015. Retrieved 29 Aug. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/exponentsandlogarithms5/graphinglogarithmicfunctions37/solvingproblemswithlogarithmicgraphs1805878/