Page:Elementary algebra (1896).djvu/376

858 ALGEBRA. Cor. If in equation (1) we put a for N, we obtain

logb a = {1 loga b} loga a = {1 loga b} logb a loga b = 1

442. Logarithms in Arithmetical Calculation. The following examples illustrate the utility of logarithms in facilitating arithmetical calculation.

Ex. 1. Given log 3 = .4771213, find log {(2.7)3 x (.81)5 + (90).

The required value = 3 log 27 + 4 log 5 — } log 90 = 8(log 33 — 1) + # (log 3! — 2) — } (log 3? + 1) logs -B+$+4) 17 = 46280766 — 5.85 = 2.780766.

The student should notice that the logarithm of 5 and its powers can always be obtained from log 2; thus

log 5 = log 10 2 = log 10 - log 2 = 1 - log 2.

Ex. 2. Find the number of digits in 87516, given log 2 = .3010300, log 7 = .8450980.

log (875^16) = 16 log (7 x 125) = 16 (log 7 + 3 log 5) = 16 (log7 + 3 — 3log2) = 16 x 2.9420080 = 47.072128;

hence the number of digits is 48. [Art. 436.]

EXAMPLES XXXVIII. a.

1. Find the logarithms of 32 and .03125 to base 3 2, and 100 and .00001 to base .01.

2. Find the value of log4 512, log 5 .0016, log81 {1 27}, log49 343.

3. Write the numbers whose logarithms to bases 25, 3, .02, 1, - 4, 1.7, 1000, are {1 2}, - 2, -3, 5, -1, 2, — {2 3} respectively.

Simplify the expressions

4. log (U2O)* 5. log : ) + : ) }. Va-3c8 aa xy? xy

6. Find by inspection the characteristics of the logarithms of 3174, 625.7, 3.502, .4, .374, .000135, 23.22065.