# Logarithms

Logarithm.

Logarithm.

*Main logarithmic identity.*

*P*

roperties of

roperties of

*logarithms. Common logarithm. Natural logarithm.*

* A logarithm of a positive number N to the base b *(

*b*> 0,

*b*1 )

*is called*

*an exponent of a power x*

, to which b must be raised to receive N.

, to which b must be raised to receive N

The designation:

&

nbsp; &n

bsp;

This record is identical to the following one:

* *

E x a m p l e s :

The above

presented definition of logarithm may be written as the *logarithmic * *identity
*:

* *

*The
main properties of logarithms.*

1)* *log * b *= 1

* , *because *b** *^{ 1}* = b .*

*b
&
nbsp; *

&

nbsp;

2) log*
*1 = 0 ,

*because*

*b*

^{0}= 1

*.*

*b*

* *

3)* Logarithm of a
product is equal to a sum of logarithms of factors:*

* *

log (* ab
*) = log

*a*+ log

*b .*

* *

4)* *

Logarithm of a quotient is equal to a

difference of logarithms of dividend

and divisor:

log (* a */* b *) = log* a *– log

b .

* *

5)* ** Logarithm of a power is equal to a product of an exponent of the power* *by logarithm of its base:*

* *

* *

* *

A consequence of this property is the following:

logarithm of a root is* equal to the logarithm of
radicand divided by the degree of the root:*

* *

6) * If a
base of logarithm is a power, then a value, reciprocal to this power exponent, may be carried out of the logarithm symbol:*

The two last properties may be united in the general

property:

&

nbsp;

* *

7)* The transition module formula ( i.e. a transition from one base of the logarithm to another base ):
*

* *

&

nbsp; &n

bsp; &nb

sp; &nbs

p;

In the particular case:* N = a *we have:* *

* *

** Common logarithm **is

a

*logarithm to the base*10. It marks as

**lg**

**, i.e. log**

_{ 10}

*N*= lg

*N*.

Logarithms of the

numbers 10,

100, 1000, … are

equal to

1, 2, 3, …

correspondingly, i.e.

they have as

many positive

ones as many zeros are placed in the number after one. Logarithms of the numbers 0.1,

0.01, 0.001, … are equal to –1, –2,

–3, …, i.e.

they have as many negative ones as

many zeros are placed in the number before one (

including zero of integer part

). Logarithms

of the rest

of the numbers have

a fractional part, called a

*mantissa*. An

integer part of logarithm is called a

*characteristics*. Common logarithms are the

most suitable for practical use.

** Natural logarithm **is a

*logarithm to the base*

*е*. It marks as

**ln**, i.e. log

_{ e}*N*= ln

*N*. The number

*е*

is irrational, its approximate value is

2.718281828459045. This number is a limit, which the number ( 1 + 1

/

*n*)

^{n}approaches at unbounded increasing of

*n*(

see

*the first remarkable limit*

on the page “Sequences. Limits of numerical sequences.

Some remarkable limits”). Strange though it may seem, natural logarithms are very

suitable at different operations in analysis of

functions. Calculation of logarithms to the base

*е*is executed quicker, than to any other base.