Before going into Natural Logarithm concept,Let's look at what is logarithm and why it is introduced. It is a fact that when we compare the operations, addition and subtraction are very easy compared to multiplication and division.
So logarithm is a method that is used to convert multiplication and division to addition and subtraction respectively. This was invented long before the use of calculators.
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The natural logarithm is the logarithm to the base ‘e'.
‘e' is an irrational number approximately equal to 2.7182.....The natural logarithms are sometimes called as the Napier Logarithms. Natural logarithm is defined for all real numbers and they can also be defined for complex numbers. E is the Euler's number.
The natural logarithm of a number x is written as loge x or simply as ln x.
Many natural phenomena like the population growth, radioactive decay, certain bacterial growth can be defined using logarithms to the base e i.e. natural logarithms. (This may one of the reasons why the logarithm to the base e are called natural logarithms)

Natural logarithms can easily be defined by using Taylor's series. Ln (b) is the area under the graph of $\frac{1}{x}$ from 0 to b. If y = ln (a) then ey = a. Hence ln(e) = 1.To understand it more clearly, Lets check the table of power of 2.

20 = 1 29 = 512
21 = 2 210 = 1024
22 = 4 211 = 2048
23 = 8 212 = 4096
24 = 16 213 = 8192
25 = 32 214 = 16384
26 = 64 215 = 32768
27 = 128 216 = 65536
28 = 256 217 = 131072

Let’s see how multiplication can be done easily.
Multiply 128 and 512.
From table 128 stands for 27
And 512 stands for 29
So 128 x 512 can be written as 27 x 29
By index rule, 27 x 29 = 2(7+9)
= 216
So the required product is the number in the table equivalent to 216 which is 65536
S0, 128 x 512= 65536
This answer was obtained without multiplication.

Division can also be done in a similar way.
Let’s take $\frac{16384}{256}$
From table 16384 stands for 214
And 256 stands for 28
So, $\frac{16384}{256}$ is same as $\frac{2^{14}}{2^{8}}$
By index rule $\frac{2^{14}}{2^{8}}$ = 2(14 - 8)
= 26
So the required product is the number in the table equivalent to 26 which is 64
So, $\frac{16384}{256}$ = 64

So, if the table was long enough to contain all powers of 2, the multiplication and division would have been a lot easy. But it has its limitations. In this term all real numbers are not present, so it’s not appropriate to use it in general
The logarithm is always considered as the inverse of power function. The logarithm or base 10 of the number is the power of 10 that gives that answer.
Taking 10 to the -3 power is same as taking the reciprocal of 103 = $\frac{1}{10^{3}}$ or 0.001. The logarithm of 0.001 is the power of 10 that would equal 0.001 which is -3.

Hence, if there was a table containing all numbers written in the exponential form of some base, then all multiplication and division could have been formed.

Lets make a new table
For that lets consider the statement 25= 32, which is written using three numbers 2, 5, and 32.
Here 2 is called BASE
5 is called EXPONENT
32 is called fifth power of 2.
Each one of these three numbers can be explained in terms of the other two as follows.
32 is the 5th POWER of the base 2
Base 2 is the 5th ROOT of the number 32.

Now how can 5 be written in terms of 2 and 32.
5 is the power to which base 2 is raised to get 32.

Hence
5 is the LOGARITHM of 32 to base 2
This is written in symbol as 5 = log2 32
This notation is known as logarithmic notation.

So, in general
Logarithm of a number N to a base x, is the exponent of base x when N is written in the exponential form. It can also be seen that the logarithm of N is the power to which the basic is raised to get the number N.

> Natural logarithm is taken with base as “e” which is an irrational constant and it takes the value 2.718281828…..

> The notation used for natural logarithm is
ln x or (loge x)


Now,
ln(e) =1, since base of natural logarithm is e.
Below are the properties of natural logarithms :

Exponential form of logarithm

As natural logarithm, ln(x) is defined as the inverse of exponential function ex, a connection can be made between them.
So, if y = ln(x)
Then its exponential form is
x = ey
So x = eln(x)
or ln(ex) = x

Example on Exponential form of logarithm
ln (x)= 5.43
By exponential form
x = e(5.43)
= 228.149 (by calculator)

Natural Log Rules

1) Natural logarithm of the product of two variables x and y is the sum of the logarithm of x and y.
ln(xy) = ln(x) + ln(y)
This is called Product rule.
For example,
ln(2x)= ln(2) + ln(x)
ln(zy) = ln(z) + ln(y)

2) Natural logarithm of the quotient of two variables x and y, is the difference of the logarithm of x and y.
ln($\frac{x}{y}$) = ln (x) - ln(y)
This is called Quotient rule.
For example,
ln($\frac{3}{y}$)= ln(3) - ln(y)
ln($\frac{5}{7}$) = ln(5) - ln(7)

3) The logarithm of the power, n of the variable x is the product of n and logarithm of x.
ln(x)n = nln(x)
This is Power rule.
For example,
ln(a)5 = 5ln(a)
ln(3)8 = 8ln(3)

Examples on Natural Log Rules


1.Expand ln(100xy)

Solution:
ln(100xy) = ln(100) + ln(x) + ln(y)
= ln(102) + ln(x) + ln(y)
= 2ln (10) + ln(x) + ln(y)

2.Expand ln ($\frac{x z^{2}}{y}$)

Solution:
ln ($\frac{x z^{2}}{y}$) = ln(x z2) – ln(y)
= ln(x) + ln(z2) – ln(y)
= ln(x) + 2 ln(z) – ln(y)

Change of Base Formula

Any base can be changed from one to another.
The rule is log a b = $\frac{ln b}{ ln a}$

Example on Change of Base Formula
Write log 8 x as natural log
Log 8 x = $\frac{ln(x)}{ln(8)}$

Another example
Log 5 7 = $\frac{ln(7)}{ln(5)}$

Properties of Natural Logarithm

  • ln(1) = 0
  • ln(e) = 1
  • ln(ex) = x
  • ln($\frac{1}{x}$) = -ln(x)
Example on Properties of Natural Logarithm

1.Expand without calculator
ln($\frac{e^{5}}{y}$)

Solution:
ln($\frac{e^{5}}{y}$) = ln(e5) – ln(y)
= 5 ln(e) – ln(y)
= 5(1) – ln(y)
= 5 – ln(y)

2.Combine $\frac{1}{3}$ ln(y) – 4ln(y) into one logarithm.
Solution:
$\frac{1}{3}$ ln(y) – 4ln(y) = ln(y)$\frac{1}{3}$ – ln(y)4
= ln($\frac{y^{\frac{1}{3}}}{y^{4}}$)

3.Solve for t
800= 3 $\times$ 7t
Solution:
Take natural logarithm on both sides
ln(800) = ln(3 $\times$ 7t)
ln(800) = ln(3) + ln(7t)
ln(800) = ln(3) + t ln(7)
So t ln(7) = ln(100) – ln(x)
t = $\frac{(ln(800) – ln(x))}{ln(7)}$
Below are the examples on solving natural logarithms with step by step solution.

1.Evaluate the following.
a) ln(e4)
b) ln (6)
c) ln($\frac{1}{e^{3}}$)

Solution:

a) By definition of natural logarithm we have,
ln(e4) = 4 ln(e) = 4

b) ln 6 = 1.7917594692280550008124773583807
This is an approximate value using the calculator.

c) ln($\frac{1}{e^{3}}$) = -ln(e3) Since ln(1/a) = -ln(a) for all a ∈ R+
= -3 ln e Since ln(ab) = b ln(a) for all a ∈ R+ and b ∈ N
= -3 Since ln e = 1

2.Simplify the following into a single logarithm
6ln(x + 3) - 7ln(x2 - 5)- $\frac{1}{2}$ ln(x+2)

Solution:
Step 1: Apply the rule, ln(ab) = b ln(a) for all a ∈ R+ and b ∈ N to all the three terms in the question.

6ln(x+3) - 7ln(x2 - 5)- $\frac{1}{2}$ ln(x + 2) = ln(x + 3)6 -ln(x2 - 5)7 - ln(x+2)$\frac{1}{2}$

Step 2: Group and Use the rule ln(ab) = ln(a) + ln(b) for all a,b ∈ R+

ln(x + 3)6 -ln(x2 - 5)7 - ln(x+2)$\frac{1}{2}$ = ln(x+3)6 - [ln(x2 - 5)7 + ln(x+2)$\frac{1}{2}$]

= ln(x+3)6 -ln{(x2 - 5)7(x+2)$\frac{1}{2}$

Step 3: Use the rule ln($\frac{a}{b}$) = ln(a) - ln(b) for all a,b ∈ R+

ln(x + 3)6 - ln{(x2 - 5)7(x+2)$\frac{1}{2}$ = ln $\frac{(x+3)^{6}}{(x^{2}-5)^{7}(x + 2)^{\frac{1}{2}}}$

Thus we have simplified.