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3. If we have to find the fourth power of 13, we proceed thus :

log1013= 1.11394

4

4.45576

which is the logarithm of 28561, the number required.

4. If we are to find the fifth root of 16807, we proceed

thus :

5)4.22549 = log1016807,

0.845098

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10

3

which is the logarithm of 7, the root required.
5. Given log102 = 0.30103; find log10128, log10512.

Ans. 2.10721, 2.70927. 6. Given log103 = 0.47712; find log1,81, log102187.

Ans. 1.90849, 3.33985. 7. Given log103; find logi 3.

V3

0.28627. 8. Find the logarithms to the base a of

a', a?", Va, Ya, a

a 3 9. Find the logarithms to the base 2 of 8, 64, 1, .125, .015625, 64.

Ans. 3, 6,

-1, – 3, – 6, 2. 10. Find the logarithms to base 4 of 8, 16, 1.5, 3.015625.

Ans. i, j, - , - 1.

Express the following logarithms in terms of log a, logb, and logc:

11. log V (a*b* c). Ans. 6 log a +9log b + 3 log c. 12. log Va: 65c".

log a + log b + Zlog c. Şab-10-2 13. log

1 log a. (a-16-4-1)

*

64. Common System of Logarithms. — There are two

systems of logarithms in use, viz., the Naperian* system and the common system.

The Naperian system is used for purely theoretic investigations; its base is e= : 2.7182818.

The common system f of logarithms is the system that is used in all practical calculations; its base is 10.

By a system of logarithms to the base 10, is meant a succession of values of x which satisfy the equation

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for all positive values of m, integral or fractional. Thus, if we suppose m to assume in succession every value from 0 to co, the corresponding values of x will form a system of logarithms, to the base 10.

Such a system is formed by means of the series of logarithms of the natural numbers from 1 to 100000, which constitute the logarithms registered in our ordinary tables.

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Also, 10-= 1o =.1, .:. log.1 = -1; 10-=ido = .01, ..

.:. log.01 = -2; 10-3= 1000= .001,

.. log.001

3. and so on.

Hence, in the common system, the logarithm of any number between

1 and 10 is some number between 0 and 1; i.e., 0+ a decimal ;

* So called from its inventor, Baron Napier, a Scotch mathematician. | First introduced in 1615 by Brigys, a contemporary of Napier.

10 and 100 is some number between 1 and 2 ; i.e., 1+ a decimal;

100 and 1000 is some number between 2 and 3; i.e., 2 + a decimal;

1 and .1 is some number between 0 and -1; i.e., -1+ a decimal ;

.1 and .01 is some number between – 1 and -2; i.e., - 2 + a decimal ;

.01 and .001 is some number between - 2 and - 3; i.e., -3+ a decimal ; and so on. It thus appears

that (1) The (common) logarithm of any number greater than 1 is positive.

(2) The logarithm of any positive number less than 1 is negative.

(3) In general, the common logarithm of a number consists of two parts, an integral part and a decimal part.

The integral part of a logarithm is called the characteristic of the logarithm, and may be either positive or negative.

The decimal part of a logarithm is called the mantissa of the logarithm, and is always kept positive.

NOTE. — It is convenient to keep the decimal part of the logarithms always posi. tive, in order that numbers consisting of the same digits in the same order may correspond to the same mantissa.

It is evident from the above examples that the characteristic of a logarithm can always be obtained by the following rule:

RULE. — The characteristic of the logarithm of a number greater than unity is one less than the number of digits in the whole number.

The characteristic of the logarithm of a number less than unity is negative, and is one more than the number of ciphers immediately after the decimal point.

a

Thus, the characteristics of the logarithms of 1234, 123.4, 1.234, 1234, .00001234, 12340, are respectively, 3, 2, 0, – 1, -5, 4.

NOTE. – When the cbaracteristic is negative, the minus sign is written over it to indicate that the characteristic alone is negative, the mantissa being always positive.

Write down the characteristics of the common logarithms of the following numbers :

1. 17601, 361.1, 4.01, 723000, 29. Ans. 4, 2, 0, 5, 1. 2. .04, .0000612, .7963, .001201, .1.

Ans. – 2, – 5, -1, – 3, – 1. 3. How many digits are there in the integral part of the numbers whose cominon logarithms are respectively 3.461, 0.30203, 5.47123, 2.67101 ?

4. Given log 2 = 0.30103; find the number of digits in the integral part of 81, 212, 162, 2100 .

Ans. 10, 4, 25, 31.

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65. Comparison of Two Systems of Logarithms. — Given the logarithm of a number to base a; to find the logarithm of the same number to base b.

Let m be any number whose logarithm to base b is required. Let

x = logom; then ht=m.

:: loga(54) = log,m; or a log, b = logam.

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Hence, to transform the logarithm of a number from base a to base b, we multiply it by

1 logab

1 This constant multiplier is called the modulus of

logąb the system of which the base is b with reference to the systein of which the base is u.

If, then, a list of logarithms to some base e can be made, we can deduce from it a list of common logarithms by multiplying each logarithm in the given list by the modulus

1 of the common system

log. 10 Putting a for m in (1), we have

1

logo a

log, a
log. b log. by (2) of Art. 63.
, b '

.

:: logo a x logab = 1.

EXAMPLES.

1. Show how to transform logarithms with base 5 to logarithms with base 125.

Let m be any number, and let x be its logarithm to base 125.

Then m = 125* = (53)* = 53. .: 3x = logs m.

.: x = log125 m = {log;m.

Thus, the logarithm of any number to base 5, divided by 3 (i.e., by logs 125), is the logarithm of the same number to the base 125.

Otherwise by the rule given in (1). Thus,

log; m log 125 m =

logsm

3

log: 125

Show how to transform

2. Logarithms with base 2 to logarithms with base 8.

Ans. Divide each logarithm by 3.

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