responding variation to b, a being constant, and is called the base of the system, and differs only in different systems.

The constant a cannot be 1, for every power of 1 is 1, and the variation of x in that case would give no variation to b; hence, the base of a system cannot be unity; in the common system it is 10.

In the equation 10x=2, x is in value a decimal fraction, and is the logarithm of the abstract number 2

In the equation a=b, if we suppose x=1, the equation becomes a'=a; that is, the logarithm of the base of any system is unity.

If we suppose x=0, the equation becomes ao =1; hence, the logarithm of 1 is 0, in every system of logarithms.

(Art. 146.) The logarithms of two or more numbers added together give the logarithm of the product of those numbers, and conversely the difference of two logarithms give the logarithm of the quotient of one number divided by the other.

For we may have the equations a=b, a=b', and a2=b". Multiply these equations together, and as we multiply powers by adding the exponents the product will be

ax+y+2bb'b"

Hence, by the definition of logarithms, x+y+z is the logarithm for the number represented by the product bb'b". Again, divide the first equation by the second, and we have

b

a=; and from these results we find that by means of a

table of logarithms multiplication may be practically performed by addition, and division by subtraction, and in this consists the great utility of logarithms.

(Art. 147.) In the equation a=b, take a=10, and x successively equal to 0, 1, 2, 3, 4, &c. Then 10°=1, 10=10, 102=100, 103=1000, &c

Therefore, for the numbers

1, 10, 100, 1000, 10000, 10000); &c., we have for corresponding logarithms

0, 1, 2, 3, 4, 5, &c.

Here it may be observed that the numbers increase in geometrical progression, and their logarithms in arithmetical progression.

Hence the number which is the geometrical mean between two given numbers must have the arithmetical mean of their logarithms, for its logarithm.

On this principle we may approximate to the logarithm of any proposed number. For example, we propose to find the logarithm of 2.

This number is between 1 and 10, and the geometrical mean between these two numbers, (Art. 122.), is 3.16227766. The arithmetical mean between 0 and 1 is 0.5; therefore, the number 3.16227766 has 0.5 for its logarithm.

Now the proposed number 2 is between 1 and 3.162, &c., and the geometrical mean between these two numbers is 1.778279, and the arithmetical mean between 0. and 0.5 is 0.25; therefore, the logarithm of 1.778279 is 0.25.

Now the proposed number 2 lies between 1.778279, and 3.16227766, and the geometrical mean between them will fall near 2, a little over, and its logarithm will be 0.375. Continuing the approximations, we may at length find the logarithm of 2 to be 0.301030, and in the same manner we may approximate to the logarithm of any other number, but the operation would be very tedious.

(Art. 148.) We may take a reverse operation, and pro-. pose a logarithm to find its corresponding number; thus, in the general equation a=n; x may be assumed, and the corresponding value of n computed.

3

3

Thus, suppose x=; then (10)=n, or 103=no,

Hence, n=1/1000=1.995262315.

That is, the number 1.9952, &c., (nearly 2) has 0.3 for its logarithm. In the same way we may compute the numbers corresponding to the logarithms 0.4, 0.5, 0.6, 0.7, &c. (Art. 149.) We may take another method of operation to find the logarithm of a number.

Let the logarithm of 3 be required.

The equation is 10=3, the object is to find 2. It is obvious that a must be a fraction, for 10'=10; therefore, 1

Let x Then 10-3, or 10-3x'. Here we perceive that x' must be a little more than 2. Make

Here we find by trial that " is between 5 and 6; take it 5, and x=2+1=1; hence, x=1=0.45, for a rough approximation for the logarithm of 3. A more exact com putation gives 0.4771213; but all these operations are exceedingly tedious, and to avoid them, mathematicians have devised a more expeditious method, by means of a converging series; but its analysis and explanation would carry us too far into analytical science to comport with the design of this work.

(Art. 150.) It is only necessary to calculate directly the logarithms of prime numbers, as the logarithms of all others may be derived from these. Thus, if we would find the logarithm of 4, we have only to double that of 2; for, taking the equation (10)=2, and squaring both members, we have, (10)2=4; or taking the same equation, and cubing both

members, we have (10)*=8; which shows that twice the logarithm of 2 is the logarithm of 4, and three times the logarithm of 2 is the logarithm of 8; and in short, the sum of two or more logarithms correspond to the logarithm of the product of their numbers. (Art. 142.)

(Art. 151.) As the logarithm of 1 is 0, 10 is 1, 100 is 2, &c., we may observe that the whole number in the logarithm is one less than the number of places in the number.

The whole number in a logarithm is called its characteristic, and is not given in the tables, as it is easily sup. plied. For example, the integral part of the logarithm of the number 67430 must be 4, as the number has 5 places. The same figures will have the same decimal part for the logarithm when a portion of them become decimal.

Thus, 67430 logarithm 4.82885

For every division by 10 of the number, we must diminish the characteristic of the logarithm by unity.

The decimal part of a logarithm is always positive; the index or characteristic becomes negative, when the number becomes less than unity.

USE AND APPLICATION OF LOGARITHMS.

(Art. 152.) The science of trigonometry, mensuration, and astronomy, can only develop the entire practical utility of logarithms. The science of algebra can only point out their nature, and the first principles on which they are founded. To explain their utility, we must suppose a table of logarithms formed, corresponding to all possible numbers, and by them we may resolve such equations as the following:

1. Given 2=10, to find the value of x.

If the two members of the equation are equal, the logarithms of the two members will be equal, therefore take the logarithm of each member; but as æ is a logarithm already, we shall have xlog. 2=log. 10. log. 10 1

Or x=

log. 2 .30103

1

=3.3219 &c.

2. Given (729)=3, to find the value of r.

Raise both members to the a power, and 3-729=93,

Or 3=36. Hence, 2-6.

4. Given ax+b=c, and a-bd, to find the values

of x and y.

By addition, 2ax=c+d. Put c+d=2m;

Then

ax-m. Take the logarithm of each mem

ber, and a log. a= log. m, or x

By subtracting the second equation from the first and

making c-d=2n, we shall find y=

3

log. m

log. a

log. n

log. a

4. Given (216)=12, to find the value of x.

Ans. x 9 log. 6

log. 12

ab-c

d

5. Given =e, to find the value of 2.

Ans.r_log.m-log.a m being equal to (de+c)

log.b

6. Given 43-16, to find the value of x. Ans. r-6.