Logarithms

An important step toward the lessening of the labour of computations was made in the seventeenth century by the discovery of logarithms. Logarithms permit us to replace long process of multiplication with simple addition; the operation of division with that of subtraction; the task of raising to any power with an easy multiplication; and extraction of any root is reduced to a single division.

The logarithm of a given number to a given base is the exponent of the power to which this base must be raised in order to obtain the given number.

Logarithms are exponents.

If a^x=b, the exponent is said to be the logarithm¹ of b to the base a, which we write x=log_ab.

The logarithm of a number to a given base is the exponent to which the base must be raised to yield the number.

Any positive number different from unity can be used as the base of a system of real logarithms.

Examples: If 10³ = 1000, then 3 = log₁₀1000

If 2³ = 8, then 3 = log₂8

If 5² = 25, then 2 = log₅25

The logarithmic system using 10 as a base is known as² the common or Griggs system and makes use of the fact that every positive number can be expressed as a power of 10. Since our number system uses 10 for a base, it is desirable for us to use 10 for the base of logarithms.

The following table shows the relationships between the exponential and logarithmic forms:

10³ = 1000 log₁₀1000 = 3.0000

10² = 100 log₁₀100 = 2.0000

10¹= 10 log₁₀10 = 1.0000

10° =1 log₁₀l = 0.0000

10^-1 = 0.l log₁₀0.1 = 1 or -1 or 9.0000-10

10^-2 = 0.01 log₁₀0.01 =2 or - 2 or 8.0000-10

From this table it is clear that any number between 100 and 1000 is a power of 10 for which the exponent is greater than 2 but less than 3, and consequently, its logarithm is between 2 and 3 (2+a decimal). Similarly, the logarithm

of 30 is (1+a decimal).

Later, when we use the table, we shall find log 30= 1.47712 which also means 10^1.47712=30. The positive decimal part of logarithm is called the mantissa and the integral part is called the characteristic.

Example. If log₁₀300=2.47712; 2 is the characteristic and 47712 is the mantissa.

Notes:

¹ the exponent is said to be the logarithm – говорят что показатель степени — это логарифм

² is known as — известна как

Exercises

I. Read the following words paying attention to the pronunciation:

coefficient, exponent, except, logarithmic, yield, application, multiplication, transformation.

II. Form adverbs and adjectives using the following suffixes and translate the newly formed words into English:

-ly: part, simple, easy, real, common, clear

-less: use, number, power

III. Answer the following questions:

1. When was the discovery of logarithms made? 2. Why was the discovery of logarithms an important step? 3 What operations can be replaced by logarithms? 4. What logarithmic system is known as the common or Griggs system? 5. What fact does the common logarithmic system make use of?

4. Translate into Russian:

In giving logarithm of a number, the base must always be specified unless it is understood from the beginning that in any discussion a certain number is to be used as base for all logarithms. Any real number except 1 may be used as

base, but we shall see later that in applications of logarithms only two bases are in common use.

Suppose the logarithm of a number in one system is known and it is desired to find the logarithm of the same number in some other system. This means that the logarithm of the number is taken with respect to two bases. It is sometimes

Important to be able to calculate one logarithms when the other in known.

5. Translate into English:

Логарифм данного числа по данному основанию – это показатель степени, в которую это основание должно быть возведено для того, чтобы получить данное число. Логарифмическая система, у которой 10 — это основание, называется общей системой или системой Григса и основывается на том факте, что каждое положительное число может быть

выражено, как степень десяти.

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