Common Logarithms.—Since ro° = r and ro'= ro, the loga rithm of I is zero and the logarithm of io is unity. Any number between r and ro has a common logarithm which lies between o and 1. Similarly, since roo, any number between ro and Ion has a common logarithm which lies between I and 2 ; and since r,000, any number between roo and r,000 has a com mon logarithm which lies between 2 and 3; and so on. Thus the logarithm of 75 is nearly 1.87506. The integral part of the log arithm is called the characteristic, the decimal part is called the mantissa. From the relations just given we obtain the general statement that the characteristic of the common logarithm of a number greater than r is one less than the number of integral figures. Thus, 3768.5 has four integral figures, hence its logarithm has the characteristic 3. From the relations ro° = •I, -or, -oor, etc., we obtain a corresponding rule for decimal fractions less than unity, and also the provision that the mantissa shall be positive ; namely the rule that the character istic of the common logarithm of a decimal fraction less than unity is negative and is numerically one more than the number of zeros immediately following the decimal point. Thus, o•000ro7 has three zeros immediately following the decimal point and the character istic of its logarithm is The minus sign is sometimes written above the characteristic, thus 4, to serve as a reminder that it applies only to the characteristic and not to the mantissa. It is simply for convenience that the mantissa, when different from zero, is always taken to be positive. This is accomplished by framing the rules for finding the characteristic, as was done above, so that, when the mantissa is not zero, the characteristic is always the integer immediately below the true value of the log arithm, and the fractional value (the mantissa) must be added to the characteristic to secure the true value of the logarithm. Thus, the logarithm of •07 lies between —1 and — 2 ; it might be taken to be — r and a minus mantissa or - 2 and a positive mantissa. The latter course is chosen, for the reason that the computation involving mantissas is simplified by taking them always positive.
The great advantage of the common system of logarithms lies in the fact that the mantissa of the same sequence of figures is the same, no matter where the decimal point is placed. Thus, 3.7568 and 375.68 have different characteristics but the same mantissa. This fact is evident from the relation which indicates that the logarithm of is less than of 375.68 by exactly 2, which is the logarithm of ioo. This property of the mantissa and the simple rules for finding the characteristic make it possible to prepare tables of common logarithms in very much more compact form than for other systems. Character istics are omitted altogether from the tables and are supplied by the computer.
Natural Logarithms.—To show how natural logarithms arise in analysis, we find the derivative of y=logbx, where b is any positive constant greater than I. We obtain y-Fh=logb(x+k), 77, p. 285). The proof as ordinarily given is to substitute in the above equation e for x and to show that this leads to an absurd ity since the left member of the equation, after being multiplied by a suitable constant C, may be broken up into two parts, one an integral part which is not zero, and the other part having a fractional value which can be made as small as we please. Evi
dently the sum of an integer different from zero and a proper fraction cannot equal zero. Hence e cannot be a root of the equation.
If in the figure we imagine all algebraic numbers marked on the n-axis and also on the 1-axes, the points so marked are "dense" on each axis i.e., between any two algebraic numbers, no matter how close they are to each other, there exists at least one other algebraic number, e.g., the arithmetic mean of the two. If in the n/-plane we imagine all points marked which have an algebraic abscissa n and also an algebraic ordinate 1, then the entire plane is "densely" covered by these algebraic points. In spite of this fact, the logarithmic curve possesses the extraordinary property of not containing any of these algebraic points, except only n = and l=o. With this exception, all points of the curve must have at least one co-ordinate which is an irrational of the transcen dental type the curve finds its way through this complicated maze of points without hitting more than once a point whose co ordinates are both algebraic. "How would Pythagoras celebrate such a discovery," exclaims Felix Klein, "if the ordinary irra tional seemed to him worthy of a hecatomb!" Change of Base.—The change of logarithms from one base to another is effected by a formula obtained as follows: From the graph here shown it is evident to the eye that positive numbers less than I have negative logarithms. Numbers very close to o have negative logarithms whose numerical values are very large. The vertical axis is an asymptote to the curve. These statements are true also for the graphs of common logarithms and of logarithms to any base b>i. It is a curious special property of the graph for natural logarithms that only one of its points has co-ordinates both of which are rational numbers, namely the point (1, o). In other words, the equation /= or el =n, can be satisfied by only one set of numbers 1 and n, of which both are real and rational, namely n = I and 1=o. This unexpected and curious property is rendered even more subtle by the addi tional fact that in all other sets of values for n and l, at least one of the two numbers is a very special type of irrational, called transcendental. As a l r e a d y stated, the base e is itself a trans cendental number. A number is called transcendental when it is not algebraic; i.e., when it cannot be the root of an algebraic equation, • • • ±fx+g=o, where m is any positive integer and the coefficients a, b, • • • , f, g are all integers, or zero, except that g cannot be zero since that would change the d gree of the equation. Algebraic numbers include all integers and rational fractions, and also all irrational numbers which are roots of algebraic equations of the kind here described. The proof that e is transcendental was a notable achievement of the French mathematician Charles Hermite, in 1873 (Comptes Rendus, vol.