LOGARITHMS.

§ 1. The logarithm of a number N is the exponent denoting the power to which a fixed number called the base must be raised in order to produce N.

Thus, if N=b", then is log, N=x; or, in words, when the base is b the logarithm of N is x.

What is the number whose logarithm is 5 when the base log, N=5; ... N=25 = 32.

is 2?

What is the number whose logarithm is 3 when the base

is 3? Ans. 27. What is the number whose logarithm is 3 when the base is 10?

What is the number whose logarithm is

base is 8?

log, N=-2;

Ans. 1000.

2 when the

What is the number whose logarithm is base is 10?

What is the number whose logarithm is base is 27?

§ 2. The logarithms of all numbers referred to the same base are said to belong to the same system of logarithms. Thus, log 106, log 107, log 1084, log 10 768, all belong to the denary or common system.

§ 3. In any system of logarithms the logarithm of the base itself is 1.

§ 4. In every system of logarithms the logarithm of 1 is 0. For

1 = 20;

1 =

=

... log21 = 0:

10°; .. log 101 = 0:

and, in general, 16°; ... log,1 = 0.

§ 5. In any system the logarithm of the reciprocal of a number is the negative of the logarithm of the number.

5

What is the number whose logarithm is base is 16?

when the

4

What is the number whose logarithm is base is 10?

16 32'

3 when the

Ans. 0.001.

§ 6. In any system the logarithm of the product of two or more numbers is equal to the sum of the logarithms of the numbers.

...log(lx m x n ) = x + y + z = logl+logm+logn.

.

It is evident that this principle may be applied to any number of factors.

What is log(8 x 4 × 32)?

8=23;

log 28 = 3:

4 = 22;

log 24 = 2:

3225; ... log,32 = 5:

log2(8 × 4 × 32) = log 28 + log24 + log232

If log 10 123 = 2.0899, what is log 1012300?

§ 7. In any system the logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator.

Proof: If l=b", mb", then is log,l=x, log,m=y;

Given: log 103 = 0.4771 and log 102 = 0.3010; what is

3

?

log 102

Ans. 0.1761.

Given: log 10 123 = 2.0899; what is log 100.123?

§ 8. In any system the logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power.

Proof: If l = b2, then is log,l=x, lm = (bx)m = bmx ;

log 7m =mx=mx logl.

Under this head may be brought a root, for a root may be regarded as a fractional power.