# 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.

#### 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=x5 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

• To estimate the value of a function between two points between which it is tabulated.

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

#### Figures

1. ##### 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.

2. ##### Points of log(y) on a Linear Scale

Notice how values of y less than 10 are indistinguibile.

3. ##### Graph of log(y) on a Semi-Log 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.

Many mathematical and physical relationships are functionally dependent on high-order variables.

Consider the Stefan-Boltzmann law, which relates the power (j*) emitted by a black body to temperature (T).

<equation contenteditable="false">$j^*= σT^4$

On a standard graph, this equation can be quite unwieldy. The fourth-degree dependence on temperature means that power increases extremely quickly. The fact that the rate is ever-increasing (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 (Figure 1).

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 0-5) 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 (Figure 3). Similar data plotted on a linear scale is less clear (Figure 2).

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 $\log{j} = 4*\log{(\sigma T)} = 4*\log{\sigma}+ 4\log{T}$. This is the equation of a straight line with a slope of $\log{T}$ and a "y intercept" of $4 \log{\sigma}$. Plotting a straight line as indicated simplifies the interpretation.

#### Key Term Glossary

constant
An identifier that is bound to an invariant value.
##### Appears in these related concepts:
degree
the sum of the exponents of a term; the order of a polynomial.
##### Appears in these related concepts:
distance
The amount of space between two points, measured along a straight line
##### Appears in these related concepts:
equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equals sign. E.g. x=5.
##### Appears in these related concepts:
function
A relation in which each element of the input is associated with exactly one element of the output.
##### Appears in these related concepts:
graph
A diagram displaying data; in particular one showing the relationship between two or more quantities, measurements or numbers.
##### Appears in these related concepts:
interpolate
To estimate the value of a function between two points between which it is tabulated.
##### Appears in these related concepts:
linear
Of or relating to a class of polynomial of the form y = ax + b .
##### Appears in these related concepts:
logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number.
##### Appears in these related concepts:
point
An entity that has a location in space or on a plane, but has no extent
##### Appears in these related concepts:
range
The set of values (points) which a function can obtain.
##### Appears in these related concepts:
rate
The relative speed of change or progress.
##### Appears in these related concepts:
sigma
The symbol Σ, used to indicate summation of a set or series.
##### Appears in these related concepts:
slope
The ratio of the vertical and horizontal distances between two points on a line; zero if the line is horizontal, undefined if it is vertical.
##### Appears in these related concepts:
variable
A symbol that represents a quantity in a mathematical expression, as used in many sciences