Logarithm. Main logarithmic identity. P
logarithms. Common logarithm. Natural logarithm.
A logarithm of a positive number N to the base b ( b > 0, b1 )
an exponent of a power x
, to which b must be raised to receive N .
This record is identical to the following one:
E x a m p l e s :
presented definition of logarithm may be written as the logarithmic identity
main properties of logarithms.
1) log b = 1
, because b 1 = b .
1 = 0 , because b0= 1 .
3) Logarithm of a
product is equal to a sum of logarithms of factors:
log ( ab
) = log a + log b .
Logarithm of a quotient is equal to a
difference of logarithms of dividend
log ( a / b ) = log a – log
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
7) The transition module formula ( i.e. a transition from one base of the logarithm to another base ):
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
100, 1000, ... are
1, 2, 3, …
they have as
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
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
å. It marks as ln , i.e. log e N = ln N . The number å
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 å is executed quicker, than to any other base.