(iii.) If the characteristic of a logarithm be +n, there are n+1 digits in the integral part of the corresponding number. (iv.) If the characteristic of a logarithm be —n, there are n−1 ciphers immediately following the decimal point of the corresponding number. 26. It has been found that 538 102.7308 to four decimal places, or log 538 2.7308. We also have = log.0538 = log-588=log 538 log 10000 = 2.7308 - 4 = .7308 - 2; log 5.38 log 38=log 538 - log 100 = 2.7308-2 = = .7308; log 53800 = log (538 x 100) = log 538 + log 100 These examples illustrate the following principle: If two numbers differ only in the position of their decimal points, their logarithms have different characteristics but the same positive mantissa. If n denote the number of places through which the decimal point has been moved in a given number a, we have since moving the decimal point a given number of places to the right or left is equivalent to multiplying or dividing by a power of 10. 27. The characteristic and the mantissa of a number less than 1 may be connected by the decimal point, if the sign (-) be written over the characteristic to indicate that the characteristic only is negative, and not the entire number. Thus instead of log .00709.8506-3=-3+.8506, we may write 3.8506; this must be distinguished from the expression -3.8506, in which the integer and the decimal are both negative. Similarly, log .0822.9138, while log 820 2.9138. = 28. The logarithms, to the base 10, of a set of consecutive positive numbers have been computed. The student is referred to Tables of Logarithms, and to works on Trigonometry, for specific directions in the use of logarithms. 29. The following relation is sometimes useful: If we have log a log b = 1. log, az and log.b=y, b2 = a (1) and a3 = b. (2) Raising (1) to the yth power, we obtain 30. If the logarithms of any system (i.e., to any base) have been calculated, the logarithms of any other system (i.e., to any other base) can be easily obtained from them. Hence, to transform the logarithm of a number from base a to base b, divide it by log, b, i.e., the logarithm of the new base to the old base. It follows that if the logarithms of any system (say, to base a) have been calculated, the logarithms of any other system (say to base b) are obtained by multiplying each logarithm of the first system by the constant 1 number The latter number is called the Modulus of the system loga b b with respect to the system a. 31. If the natural logarithm of a number N be denoted by n and its common logarithm by m, then The value of log10 2.7182818... is known to be .4342945, to seven places of decimals. The number 2.3025851 is therefore the modulus of the natural system with respect to the common system of logarithms, i.e., is the number by which each logarithm of the common system must be multiplied in order to give the logarithms of the natural system. or, 4342945 is the modulus of the common system with respect to the natural system of logarithms. EXERCISES III. Given log 2 = .3010, log 3.4771, log 7 = .8451, find the logarithms of the following numbers: Determine the number of integral places in the following powers: 11. 2100. 12. 760. 13. 51000. 14. 294999. Exponential and Logarithmic Equations. 32. An Exponential Equation is an equation in which the unknown number appears as an exponent of a known or an unknown number; as, ax = b. A Logarithmic Equation is an equation in which the logarithm of the unknown number, or of an expression containing the unknown number, enters; as, log (x + 1) = 2. The solutions of certain forms of exponential and logarithmic equations will be considered. This result could have been obtained by inspection, by writing 3* = 32. Ex. 2. Solve the equation 3x = 5. Ex. 5. Solve the equation 10.2 - 22x = 16. Since 22 = (2x)2, we solve the equation as an equation in 2′′ as the unknown number. Replacing 2 by y, we obtain This example could also have been solved by inspection. Assuming tentatively x = 1 and y = 2 in equation (5), we see that these values also satisfy equation (6). |