Exercises.-(These exercises are of the highest importance.) Prove each of these results from the d. c.'s of sin x and cos x on the principles explained in Chapter V. in the following

Prove also the following results by the same method as that explained for y = sin1x, drawing the curve in each

CHAPTER VII.

DIFFERENTIAL COEFFICIENTS OF LOGARITHMIC FUNCTIONS.

$ 40. D.C. OF LOG x.

We will now consider the curve y = log x. A remark similar to the one we made in defining the meaning of such expressions as sin x applies here, viz. that in abstract mathematics log x with no suffix signifies, not the ordinary logarithm as found in log tables, but the "natural" logarithm to base "e" where is the value of the infinite series

I

e

which is the value which the expression (1+

n

n

assumes

when is infinitely great. The student cannot hope to understand this fully unless he be acquainted with the algebraical theory of logarithms, which is found in any fairly advanced book on algebra, such as Hall and Knights' "Higher Algebra." He may, nevertheless, obtain approximate values of the natural logarithm of a number by multiplying its ordinary logarithm (to base 10) by the log of 10 to base e, viz. 2.303 about.

Calculate in this way the natural logarithm of o*25, 0'5, 0'75, 125, 150, 20, 30, 40, 50, 75, 10. Plot points whose abscissæ are the numbers here given in inches or other units, ar ordinates the calculated logarithms. Carefully draw a smooth curve through these points. This curve crosses the line OX at

a point whose abscissa is 1; for with any base whatever log I = 0.

On the left of point (1,0) care must be taken: thus from the tables we can find log10 0'5 = 69897, which for our

purpose is practically equivalent to o'700, since we cannot

Draw a curve through the points. This curve is shown in a full line in Fig. 38.

Now differentiate this curve graphically. The general shape of the curve obtained will be as shown in the lower part of Fig. 38. Take a number of points such as P' on the curve, measure with a decimal scale p'P' and O'p', multiply their lengths together, and the result will be found to be always if the work is accurately done. Its equation

Another way of exhibiting this fact very clearly is to take a number of points P on the primary, through which erect ¿T perpendicular = 1 inch. Join OT. Then OT will be found parallel to the tangent PR at P.

Exercise. The whole curve may therefore be drawn by the method explained in § 14. Draw it in this way, and compare it with the curve just plotted.

These exceedingly important facts may be proved algebraically as follows. Consider another ordinate qQ near pP, distance h from it.