b

10 as a standard value, and we shall now show that we can determine a constant m such that the graphs of kb and k10mx are identical.

The graphs will be identical if and only if

This is essentially an exponential equation to be solved for m. Equating the logarithms of both sides of the equation,

Hence it is always possible to use the base 10, and we shall do so in the future unless the contrary is indicated.

The most characteristic properties of the graph of y = k10mx are that it does not cross the x-axis, that it does cross the y-axis, and that it always rises or always falls.

obtained from it? This graph is called the probability curve.

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3. Find the inverse of the function k10mx. What are the most im

portant characteristics of the graph of y = c log nx?

4. (a) Construct the graph of k log2 x for several values of k.

(b) Construct the graph of log2 nx for several values of n.

(c) Construct the graph of log2 3x.

89. The Logarithmic Scale.

The ordinates of points on the graph of log10 x corresponding to integral values of x from 1 to 10

inclusive, are a set of segments on the y-axis whose lengths from the origin are equal to the logarithms of the corresponding abscissas.

The line O'A' is an enlargement of the segment OA on the y-axis, the numbers on the right being the logarithms of those on the left.

A 10-0-1.00

10-9-1

-0.95

80.90

-0.85

-0.78

5--0.70

A scale is called a uniform scale if the distance of a number from the point marked 0 is equal to the number, the distance from 0 to 1 being the unit segment.

A scale is called a logarithmic scale if the distance of a number from the point marked 1 is equal to the 9.00 logarithm of the number, the distance from 1 to 10 being taken as unity.

8--0.48

2-8-0.30

00

The logarithms are spaced uniformly along the line O'A', while the integers are spaced non-uniformly.

The utility of a logarithmic scale lies in the fact that the addition and subtraction of the logarithms. of numbers, and hence the multiplication and division of numbers, may be effected mechanically by the addition and subtraction of line segments on the FIG. 144. scale. For instance, to find the product 3 x 2, we add the segment 12 to the segment 13. For this purpose, place a pair of dividers so that the points are on the extremities of the segment 12, and then with this opening place one end of the dividers on the point 3 and the other will touch the scale exterior to the segment 1 - 3 in the point 6, the required product.

To find the quotient, subtract the segment 12 from the segment 1-3 in a similar manner by means of the dividers.

The logarithmic scale is sometimes called Gunter's scale after Edmund Gunter, who first made use of it for purposes of calculation in 1620.

The dividers may be dispensed with if two identical logarithmic scales are arranged to slide along one another as shown in Fig. 145, which shows the method of finding the product 3 × 2 and the quotient 3/2.

For the product 4 × 5, the above method gives a point out

side the scale. To avoid this we divide one of the factors by 10, then multiply by the other factor and move the decimal point one place to the right in the result.

To find the quotient %, divide 10 by 5, multiply by 4 and move the decimal point one place to the left in the result. The following figures illustrate the methods:

The logarithmic scale finds practical application in the slide rule, an instrument much used by engineers in calculations.

The slide rule consists of a ruler with a central portion which slides back and forth in grooves on the rule, the slide and rule

being graduated with equal logarithmic scales which are labeled with the corresponding numbers.

The ordinary slide rule has four scales A, B, C, D, as shown

in the figure.

Each of the scales C and D is a single logarithmic scale. Each of the scales A and B consists of two logarithmic scales, which are equal.

If we give to the left index on all four scales the value 1, then the scales A and B extend from 1 to 100 while scales

C and D extend from 1 to 10. Hence the scales A and B, numerically considered, are twice as long as scales C and D, and a number on the upper scales is the square of the number below it on the lower scales. Conversely, a number on the lower scales is the square root of the number directly above it on the upper scales.

The methods employed to keep the result on the scales in calculations of the type 4 x 5 and 4/5 are not necessary if the upper scales are used. The lower scales give more accurate results, since the graduations are not so fine as those of the upper scales.

The instrument is provided with a runner, R, which enables coinciding points to be found on any of the scales, and also permits of extensive calculations being worked out without the necessity of recording the intermediate results.

The slide rule makes use of matissas only, the characteristics being determined by inspection.

Logarithmic scales are also used on the axes of cross-section paper. If the scales on both axes are logarithmic the crosssection paper is called logarithmic paper. If the scale on one of the axes is logarithmic and on the other is uniform it is called semi-logarithmic paper. The methods of using these crosssection papers are shown in the following examples.

EXAMPLE 1. Plot the graph of the equation obtained by taking logarithms of both sides of the equation y = 3x2.

The required equation is log y

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12

B

Letting log y = Y and log x = X, we have Y = 2X + log 3, which is a linear equation, and hence its graph is the straight line whose slope is 2 and whose intercept on the Y-axis is log 3. The tables give corresponding pairs of values of x, y and of X, Y, and the figures show corresponding parts of the graphs. Only the part of the graph of Y 3x2 in the first quadrant corresponds to the graph in the X, Y system, as the logarithm of a negative number is not a real number.

bers on the uniform scales attached to the figure, so that X = log x and Y = log y.

We follow the same procedure as in plotting the graph of the equation on ordinary cross-section paper, except that negative values of x and y are not used, and the lines through the point (1, 1) are usually chosen as the axes since log 1 0.