What is Natural Logarithm and How to Calculate it?

Introduction to Natural Log

A very conceptual mathematical topic, natural logarithm is a bit complex yet interesting. The natural log of a number is defined as its logarithm to the base of the mathematical constant e. The constant e is an irrational and transcendental number, which has a value equal to 2.718281828459.

Therefore, if we try to find the natural logarithm of any variable, namely x, it will be written as log_e x or simply log x. In simpler words, the natural logarithm of x is the power to which e would have to be raised to equal x.

Natural Log Rules

There are a large number of logarithmic rules. However, the major ones are listed below:

Logarithm product rule: The logarithm of the multiplication of x and y is the sum of the logarithm of x and logarithm of y. [log_b(x ∙ y) = log_b(x)+ log_b(y)]

For example: log₁₀(4 ∙ 8) = log₁₀(4) + log₁₀(8)

Logarithm quotient rule: The logarithm of the division of x and y provides us with the difference of logarithm of x and logarithm of y. [log_b(x / y) = log_b(x)- log_b(y)]

For example: log₁₀(4 / 8) = log₁₀(4) – log₁₀(8)

Logarithm power rule: The logarithm of x raised to the power of y gives us y times the logarithm of x. [log_b(x ^y) = y ∙ log_b(x)]

For example: log₁₀(3⁸) = 8∙ log₁₀(3)

Logarithm base switch rule: The logarithm of x to the base y is equal to 1 divided by logarithm of y to the base x. [log_b(x) = 1 / log_c(y)]

For example: log₃(4) = 1 / log₄(3)

Logarithm base change rule: The logarithm of x to the base y is equal to the logarithm of x to the base z divided by the logarithm of y to the base z minus logarithm of x. [log_y(x) = log_z(x) / log_c(b) Logarithm – log(x)]

For example: log₄(3) = log₅(3) / log₅(4) Logarithm – log(3)

Now when you have understood all the above natural logarithmic properties, let us uncover its calculation tactics.

How to calculate Natural Log?

Before the invention of scientific calculators and computers, the natural log was calculated using the logarithmic or log tables. However, these tables are still in use among the students during their exams. Not just that, these tables also help in calculating or multiplying large numbers. Therefore, to calculate natural log using the log table, follow the below-listed steps:

Step 1: Choose the appropriate logarithmic table according to the base. Usually, most of the log tables are for base-10 logarithms, also known as “common logs.” For example, log10(31.62) requires a base-10 table.

Step 2: Look for the precise cell value at the following intersections and ignoring all the decimal places. Consider the row labelled with the first two digits of n and column header with the third digit of n. For example, log₁₀(31.62) → row 31, column 6 → cell value 0.4997.

Step 3: Use these to adjust the answer if n has four or more significant digits. While staying in the same row, find a small column header with the fourth digits of n and add it to the previous value. In the above example of log₁₀(31.62) → row 31, small column 2 → cell value 2 → 4997 + 2 = 4999.

Step 4: Further, prefix a decimal point also known as “mantissa.” Solution to the above example so far is ?.4999

Step 5: Finally, find the integer portion which is also called as “characteristic” through trial and error method, therefore find the integer value of p such that, a^p < n and a^p+1 > n {displaystyle a^{p+1}>n}. Example: 10¹ = 10 < 31.62 and 10² = 100 > 31.62 {displaystyle 10^{2}=100>31.62}. The “characteristic” is 1.

Therefore, the final answer is 1.4999

Solved Question for You

Question: Which of the following is not a property of natural log

(a) Logarithm product rule

(b) Logarithm power rule

(d) Logarithm base switch rule

Answer: The correct answer is option (c) Logarithm numerator rule. This because there is no such property of natural log as Logarithm numerator rule.

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