The power to which a number should be raised to get the specified number is called the logarithm of that number. For example, logarithm to the base 10 of 1000 is 3 because 10 raised to the power 3 is 1000. Logarithmic function is the inverse Mathematical function of exponential function. The logarithmic function log ax = y is equal to x = ay.

There are two types of logarithms generally used in Mathematics. They are common logarithms and natural logarithms.

Common logarithm is any logarithmic function with base 10. It is generally represented as y = log x or y = log 10x.

Natural logarithms are the logarithmic functions which have the base equal to ‘e’. Natural logarithms are generally represented as y = log ex or y = ln x . ‘e’ is an irrational constant used in many Mathematical Calculations. The value of ‘e’ is 2.718281828…

The number ‘e’ is an irrational Mathematical constant and is used as the base of natural logarithms. The number ‘e’ is the only unique number whose value of natural logarithm is equal to unity. The value of ‘e’ was calculated in 1683 by Jacob Bernoulli. This mathematical constant finds its importance in various fields of Mathematics including:

Compound interest

Bernoulli trials

Standard normal distribution

Derangements

Optimal planning problems

Asymptotics

Calculus

Value of log e can be calculated in two different cases. The two cases are finding the natural logarithmic value of e and the common logarithmic value of ‘e’.

Case 1: Value of Log e to the Base ‘e’ (Natural Logarithm of ‘e’)

By definition, any logarithmic function in the inverse function of exponential function.

So if log ee = y, it can be written as e = ey.

log ee = y

e = ey

Since the bases of the exponential functions on both sides are the same, powers should also be identical according to the properties of exponential functions. Therefore, it can be inferred that the value of ‘y’ is equal to one.

Initially it was assumed that log ee = y. So we can conclude that log ee = 1. Natural logarithm of ‘e’ is equal to unity.

Case 2: What is the Value of Log e Base 10 (Common Logarithm of ‘e’):

It is a fact that the common logarithm of a function whose natural logarithm value is known can be determined by dividing the value of natural logarithm by 2.303. (The natural logarithm of any function is divided by 2.303 to obtain the common logarithmic value because the natural logarithm of 10 i.e. log 10 base e is calculated as 2.303).

In case 1, the value of log e to the base e calculated is 1. Log e base 10 is obtained by dividing 1 by 2.303.

\[log_{10}x = \frac{\text{ln x}}{2.303}\]

\[log_{10}e = \frac{\text{ln e}}{2.303}\]

\[log_{10}e = \frac{1}{2.303}\]

\[log_{10}e = 0.43421\]

Therefore the value of log e base 10 is equal to 0.43421 up to five decimal places.

Property 1: Product Rule

The natural logarithm of a product of two numbers is equal to the sum of natural logarithms of individual numbers.

\[ln{\text{ }}\left( {x.y} \right){\text{ }} = {\text{ }}ln{\text{ }}\left( x \right){\text{ }} + {\text{ }}ln{\text{ }}\left( y \right)\]

Property 2: Quotient Rule

Natural logarithm of fraction of two numbers is equal to the subtraction of natural logarithm of denominator from the natural logarithm of numerator.

\[ln\frac{x}{y} = {\text{ }}ln{\text{ }}\left( x \right){\text{ }} - {\text{ }}ln{\text{ }}\left( y \right)\]

Property 3: Power Rule

Natural logarithm of a number raised to the power of another number is equal to the product of the power and the natural logarithm of the base.

\[ln{\text{ }}\left( {{x^y}} \right){\text{ }} = {\text{ }}y{\text{ }}ln{\text{ }}x\]

Property 4: Derivative of Natural Logarithm

The first order derivative of a natural logarithmic function is equal to the reciprocal of that function.

If y = ln (x) then y’ = \[\frac{1}{x}\]

Property 5:

Natural logarithm of any number less than zero (negative numbers) is undefined.

Property 6:

Natural logarithm of 1 is equal to zero.

Property 7:

Natural logarithm of infinity is infinity.

Property 8:

Natural logarithm of -1 is a constant known as Euler’s constant.

ln (-1) = iπ (Euler’s Constant)

1. Evaluate the value of natural logarithm of √e

Solution:

ln √e = ln e½

= ½ ln e (Power rule of logarithmic function)

= ½ x 1 (Natural value of log e is unity)

= ½

2. Evaluate ln \[\frac{{\sqrt {x - 1} }}{e}\]

3. Solution:

\[ln \frac{\sqrt{x - 1}}{e} = ln \frac{(x - 1)^{½}}{e}\]

= ½ ln (x - 1) - ln e (Power rule and quotient rule)

= ½ ln (x - 1) - 1 (Recall what is the value of log e to the base e)

Fun Facts:

Derivative of the natural logarithm of ‘e’ is equal to zero because the value of log e to the base e is equal to one which is a constant value. The derivative of any constant value is equal to zero.

The logarithmic value of any number is equal to one when the base is equal to the number whose log is to be determined. Example: Log e base e is equal to 1 whereas log 10 base e is not equal to one.

Common logarithm of one is equal to zero.

The value of log 10 base e is equal to 2.303.

FAQ (Frequently Asked Questions)

1. What is the relation between logarithmic functions and exponential functions?

An exponential function is a function in which a number or a variable is raised to the power of another number or a variable. The number whose power is raised is called the base and the number to which the base is raised is called the power or index. In general an exponential function y = a^{x} means that the value of y is equal to the value of a multiplied by itself (x - 1) times. In contrary, logarithmic function is the inverse of exponential function. Logarithmic function of any number is the power to which the number must be raised in order to obtain a specific value. In general, the relationship between exponential function and logarithmic function is Mathematically represented as shown below.

Y = log _{a}X can be exponentially represented as X = a^{y}.

2. Where are logarithmic functions used in real life?

The word logarithm was coined by a Scottish Mathematician John Napier. He derived the word logarithm from Gree words Logo which means ratio and arithmos which means numbers. Logarithmic functions are mainly used in population growth, carbon dating and investment. Logarithmic functions are also used in decibel measure of sound, measuring the magnitude of earthquakes, estimation of brightness of stars, measure of pH, acidity and alkalinity etc.