Natural Logarithms Learning

Let a, b be two positive real numbers and a≠1. The real number x such that ax =b is called logarithm of b to the base a. It is denoted by logab.

Thus the logarithm of a number to the given base is the index or the power to which the base should be raised to get the given number. Logarithms  are defined only for positive real numbers. Further, there exists only a unique x which satisfies ax  =b.

Between, if you have problem on these topics Fraction and Decimal Equivalents, please browse expert math related websites for more help on neet 2013 syllabus.

Thus logab is unique.

1)If a is a positive real number and a≠1 then logaa=1.

2)If a is a positive real number and a≠1, then loga1=0.

3)If a,m are positive real numbers and a≠1, then a logam = m

4)If a, m,n are positive real numbers and a≠1 then loga(mn)=logam+logan

5)If a,m,n are real positive real numbers and a≠1, then loga(m/n) =logam-logan

6)If a,m are positive real numbers, a≠1, n is a real number, then loga(mn)=nlogam.

Functions defined by such equations are called logarithmic functions. A logarithmic function has its domain as the set of positive real numbers and its range as the set of all real numbers.

Examples to learn logarithms :

Ex1: 16=24 =>log216 =4

Ex2: 125=53 =>log5125 =3

Hint1:The logarithms of the same number to different bases are different.

Ex1: 64=26 =>log264 = 6

Ex2: 64=43 =>log464= 3

Hint2: The logarithm of 1 to any base is zero. Since 1=a0

The logarithm of a number (≠0) to the same base is unity.

since,a=a1,logaa=1

Note:logax = loga y <=>x=y;

logax ≠ loga y  <=>x≠y.

Let       logax=n    therefore x=an         => x=aloga x

alogax  =x, ,is called the fundamental logarithmic identity



Definitions of natural logarithms learning

Common logarithms:- The logarithms which are calculated to the base e (an irrational number which is approximately 2.7182...) are called natural logarithms or Naperian logarithms. logen is denoted by ln n.

The logarithms which are calculated to the base 10 are called common logarithms or Briggs logarithms. Log10n is simply denoted by log n

If n is a positive real number, then the integer k such that k≤logn
Solve the problems on natural logarithms learning

Q: 1 Find the value of log √7343

Sol:- Given log √7343

log 71/2  73

=3/(1/2) log77 = 6

Q :2  If (3.7)x = (0.037)x = 1000 then find the value of (1/x)-(1/y).

Sol :- Given (3.7)x = (0.037)y = 1000

log10(3.7)x = log10 (0.037)y =log10103

xlog10(3.7)=ylog10 (0.037)=3

log10(3.7)=3/x,   log10 (0.037)=3/y

3/x-3/y = log10(3.7) - log10 (0.037)

log10(3.7/0.037) = 2

1/x-1/y = 2/3.