153. Exponential Equations. When the unknown quantity occurs in an equation as an exponent, the equation is called an exponential equation.

The test is especially important because the division in step (4) is not exact. The approximation is so close, however, that the correct number will always be suggested, and the test will finally verify it.

It should also be noted that step (4) contains the quotient of two logarithms, which is not the same as the logarithm of the quotient; hence, Sec. 150, II, is not applicable.

13. Solve for x and y, computing the values to 2 decimal

17. It is known that the volume of a sphere is, being the length of the radius. Using 3.14 as the approximate value of π, find by logarithms the volume of a sphere of radius 7.3 in.

18. Find, as above, the volume of a sphere whose radius is 36.4 ft.

29. Given a = 0.4916, c = 0.7544, and b= c2- a2. Find b. SUGGESTION. b = (c− a) (c + a).

30. It is known that in steam engines, the piston head's average velocity (c) per second is approximately given by the formula:

where s denotes the distance over which the piston moves (expressed in the same unit as c), and p the number of pounds pressure in the cylinder.

(1) Find

C, if s = 32.5 in., p = 110 lb.

(2) Find P, if c= 15 ft., s = 2.6 ft.

31. Solve for x and y, computing decimal results to 2 decimal places:

The following questions summarize the definitions and processes treated in this chapter:

1. What is a logarithm?

2. What is the meaning of base in logarithms?

3. What is the integral part of a logarithm called? is the decimal part called?

4. State how to find the logarithm of a product.

5. State how to find the logarithm of a quotient.

Sec. 142.

Sec. 143.

What Sec. 144.

Sec. 150, I.

Sec. 150, II.

6. How is the logarithm of a power found?

Sec 150, III.

7. State each of the processes named in statements 4, 5, and 6 in the form of laws of exponents. Sec. 150.

HISTORICAL NOTE

It is remarkable that the discovery of Logarithms occurred so timely, for this great machine of computation was invented by Napier at the beginning of the seventeenth century, just in time to aid the new work in astronomy and navigation; Galileo had devised the telescope and Kepler was ready to calculate the orbits of the planets. It is also remarkable that Napier worked out the principle of Logarithms without the use of exponents, and this peculiar method, which is too complex to be explained here, produced tables quite different from those now in use. As soon as Napier's great work on Logarithms was published, Henry Briggs, a teacher in Gresham College, London, hastened to visit Napier, and suggested the advantages of the base 10, and thus laid the foundation of our tables of common Logarithms.

John Napier, also known as the Baron of Merchiston, was a Scotchman, born in 1550, and published his work on Logarithms in 1614, only three years before his death. The modesty and simplicity of the great Scottish philosopher is shown by his attitude toward Briggs, for when informed that the latter had been obliged to postpone his promised visit, Napier regretfully replied, "Ah, Mr. Briggs will not come."

JOHN NAPIER

Napier's discovery, whose importance is not exaggerated by the claim that "it doubled the life of the astronomer by shortening his labor," followed immediately upon the general acceptance of the Hindoo notation and the introduction by Stevin of decimal fractions. Thus, the seventeenth century saw the perfection of the

decimal fractions, and Logarithms.

CHAPTER V

IMAGINARY AND COMPLEX NUMBERS

154. Imaginary Numbers. The numbers defined in what precedes have all had positive squares. Consequently, among them the equation 23, which asks, "What is the number whose square is - 3?" has no solution.

A solution is provided by defining a new number, -3, as a number whose square is -3. Similarly we define √—a, a being a positive number, as a number whose square is -α. The square roots of negative numbers are called imaginary numbers.

155. If a is positive, √—a may be expressed √ √−1.

156. Real Numbers. In distinction from imaginary numbers, the numbers hitherto studied are called real numbers.

157. The positive square root of 1 is frequently denoted by the symbol i; that is, V-1=i.

Using this we write :

√ = 5 = √5. i;

49 =7i; also v 75 a2b= 5 a√3 b. i.

NOTE. Throughout this chapter the radical sign is taken as positive.