Exponential and logarithmic functions

Exponential and logarithmic functions

exponential function

exponential function is defined by the equation:

where the base a is a given positive number, 

different from unity. The value of ax is evident for integral positive x. For fractional positive x, a^x is defined as the radical where, with even q, we agree to take the positive value of the radical. Without entering at present into a detailed consideration of the value of a^x. for irrational x, it will simply be said that approximate values of a^x are obtained by taking approximate values for the irrational x (as described in types of numbers); and the closer the approximation for x, the closer for a^x, For instance, as approximations 

Evaluation of a^x for negative x follows from evaluation for positive x, since . Since we agreed above always to take the positive radical in the expression , it follows that the function a^x is positive for all real x. Apart from this, it may be mentioned in passing that it can be shown that a^x is an increasing function for a > 1, and a decreasing function for 0 < a < 1.

 Figure 29 illustrates the graph of function (15) for various values of a. We shall note some peculiarities of these graphs. We have, first of all, a^0= 1 for any a, so that the graph of function (15) passes through the point y = 1 on the axis Y for any a, i.e.

through the point with coordinates x=0, y = 1. For a > 1, the curve rises indefinitely from left to right (in the direction of increasing x). On moving to the left, it approaches the axis X indefinitely, without ever touching it. The curve is differently situated relative to the axes for a < 1. On moving to the right, the curve indefinitely approaches the axis X, whilst it rises indefinitely on moving to the left. Since a^x is always positive, the graph is evidently located above the axis X, It may be noted further, that the graph of the function y=(1/a)^x can be obtained from the graph of y = a^x, by turning the figure through 180° about the axis Y . This follows directly from the fact that, with such turning, x becomes -x, and a^-x = (1/a)^x .

A further remark: if a = 1, then y = 1^x, so that we have y = 1 for all x.

logarithmic function

A logarithmic function is defined by the equation:

By definition of logarithm, function (16) is the inverse offunction (15). Thus, we can obtain the graph of the logarithmic function (Fig. 30) from the graph of the exponential, by turning the curves of Fig. 29 through 180° about the bisector of the first quadrant of the axes.

 Since function (15) is increasing for a > 1, the inverse function (16) is also increasing and single-valued, being only defined for x > 0, as is evident from Fig. 29 (logarithms of negative numbers do not exist). All the graphs of Fig. 30 cut the axis X in the point x=1. This corresponds to the fact that the logarithm of unity is zero for any base. For the sake of clarity, Fig. 31 shows a single graph of (16) for a > 1.

The concept of logarithmic scale and the theory of the logarithmic slide-rule are closely associated with the concept of logarithmic function. A scale is called logarithmic when it is drawn on a given line so that the length of a division, instead of corresponding with the number which denotes the division, corresponds with the logarithm of this number, usually to base 10 (Fig. 32).

Thus, if a certain division of the scale denotes the number x, the length of the segment 1x is equal to , instead of x. The length of the segment between two points of the scale, denoting x and y, will be:

the logarithm of the product x, y is obtained simply by adding segment 1y to 1x, since the segment thus obtained is equal to:

Thus, given a logarithmic scale, multiplication and division of numbers can be carried out simply by adding and subtracting segments of the scale, this being realized most simply in practice with the aid of two identical scales, one of which can slide along the other (Figs. 32 and 33). This is the idea underlying the construction of logarithmic slide-rules.

Logarithmic paper is often used for calculations; this consists of ruled sheets, where the divisions of the axes X and Y are in accordance with a logarithmic, instead of an ordinary, scale.

the source:

A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek