Logarithms. The most important patterns

Logarithms were discovered 400 years ago and were widely used until the 1980s. What is the definition of logarithms? What are the laws of operations of logarithms? What is a slide rule?

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1. Logarithm. Definition

We call the logarithm of b at base a c such that a is raised to the power of c. In the language of mathematics, this definition can be expressed as follows:

logab = c ↔ac = b

So the logarithm is the inverse of exponentiation.

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2. Logarithms. Discovery

Logarithms were discovered in the 16th century. They were developed by the Scottish mathematician and aristocrat Ioannes Neper and the English mathematician and astronomer Henry Briggs.

In those days, astronomy, on which navigation and trade depended, required tedious calculations on paper. The discovery of logarithms made it possible to replace multiplication, division and square root with easier addition, subtraction and division by a natural number.

After first Briggs and then Neper's works were published, log tables and sliders became widely used in scientific, engineering, and astronomical calculations.

3. Logarithms. The basics

The base or base of the logarithm is called the number a. In turn, b is a logarithm which can also be the antilog of its logarithm. So this is the exponent of the power to which the base a should be raised to get the logarithm of b.

Here is an example:

Log2 8 = 3 as 23 = 8

The logarithm must satisfy three conditions, which are also called assumptions or the log domain:

the base of the logarithm must always be a positive number, that is: a> 0,

the base is not 1, therefore: a ≠ 1,

the logarithm must be positive, that is: b> 0.

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4. Logarithms. The most important patterns

Formula for adding and subtracting logarithms with the same base:

logab + logac = logs (b⋅c) logab − logac = loga (b¦c)

Taking the exponent before the logarithm:

loga (bn) = n ⋅ logab loganb = 1 / n logab

Logarithm in the exponent of the power: alogab = b

5. Logarithms. Action rights

Here are the main assumptions:

a> 0, a ≠ 1, b> 0, x> 0, y> 0

Product logarithm:

loga (x ⋅ y) = logax + logx + logay

Quotient logarithm:

logs x / y = logax - logay Power logarithm: xy logos = y x logos root logarithm: logs √ (n & x) = 1 / n logax

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6. Natural logarithm

The natural logarithm is also called the Neper logarithm, which used logarithms close to 1 / e;

The number e, i.e. the Euler number, can be defined as the limit of a certain numerical sequence. The number e is approximately 2.718281828.

7. Natural logarithm

The decimal logarithm is also called Briggs because it was introduced in 1614 by Henry Briggs.

The logarithm of base 10 consists of:

  • integral part, called a feature;
  • a decimal point, called the mantissa.

The decimal logarithm is determined as follows:

Lg x = log10 x

The feature of the logarithm of the number x (for x ≥ 1) is smaller by one than the number of digits before the decimal point in the notation of the number x.

8. Logarithmic slider

The logarithmic slider or calculator slider can be called the predecessor of the calculator. It was invented in 1632 by the English mathematician William Oughtred.

The logarithmic slider works by adding logarithms by adding different length segments marked on the scale:

log (a⋅b) = log (a) + log (b)

The slider made the calculations much easier and was used by engineers, physicists and mathematicians until the end of the 1980s.

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