Fundamentals of Logarithms

Fundamentals of Logarithms A logarithm calculates how many times a base multiplies itself to get to another number. Base in logs refers to the multiplied number. Logarithms are another way to express exponents in mathematics. Logarithms are the best way to express large values. In Greek, Logaritimos is another word for logarithms.

Background of Logarithms

In the 17th century, John Napier introduced the idea of logarithms. He used the Greek words logos and arithmos. Logarithms were essential for navigation, astronomy, surveying, and engineering calculations. These are the inverse practice of exponentiation.


John Napier introduced the idea of logarithms in the 17th century.

Formula of Logarithms

The formula is as follows:


c^z = a ⇔ log c a = z

  • In the above equation, c^z = a is the exponential part.
  • In this case, “log” refers to a logarithm.
  • Both “a” and “c” are positive real numbers.
  • A number “z” is a real number.
  • Inside log “a” is the argument.
  • Bottom of the log contains c as a base.

Types of Logarithms

The types of logarithms are as follows:

1. Common Logarithm

The base ten logarithms are another name for the common logarithm. It’s written as log10 or log. The common logarithm tells the number of times we must multiply by 10 to get the desired output. For illustration, log (100) = 2.The result of multiplying ten by itself twice is 100. The formula for the common log is as follows:

  • log 10 = log.

Example of Common Logarithm

  • 10^2 = 100.
    = log 100 = 2.

2. Natural Logarithm

The base e logarithm is the natural logarithm. Ln or log are symbols for the natural logarithm. The Euler’s constant, “e,” is roughly equal to 2.71828. For example, ln 78 is the natural logarithm of 78. The natural logarithm indicates how many times we must multiply “e” to get the desired result. For illustration, ln (78) = 4.357. As a result, 78’s base e logarithm is equal to 4.357. The formula for the natural log is as follows:

  • loge = ln.

Example of Natural Logarithm

  • e^x=2.
    = ln 2 = x.


The two types of logarithms are common logs and natural logs.

Types of Logarithms

Why do we need Logarithms?

The need for logarithms is as follows:

  1. Logarithms describe multiplication-based changes.
  2. Logarithms assist in calculating a series of multiplications.
  3. Logarithms are mathematical relationships employed in scale comparisons.

Other formulas of Logarithms

The properties are as follows:

  • log b(mn) = log b(m) + log b(n).
  • m log b(x) + n log b(y) = log b(xmyn).
  • log b(m/n) = log b (m) – log b (n).
  • log b (xy) = y log b(x).
  • log bm√n = log b n/m.
  • log b(m+n) = log b m + log b(1+nm).
  • log b(m – n) = log b m + log b (1-n/m).

Rules of Logarithms

The rules are as follows:

1. Product rule

Multiplying two logarithmic numbers is equal to adding their logarithms. The formula is as follows:

  • Log b(mn)= log b m + log b n.

2. Exponential rule

The exponent times the logarithm of m’s logarithm is equal to m’s logarithm with a rational exponent. The formula is as follows:

  • Log b(mn) = n log b m.

3. Division rule

The difference between each logarithm equals the division of two logarithmic values.

  • Log b(m/n)= log b m – log b n.

4. Derivative of log rule

In this rule, if f (x) = log b (x), then f(x)'s derivative is;

  • f '(x)=1/(x ln(b)).

5. Integral of log rule

The rule is as follows:

  • ∫log b(x)dx = x( logb(x) – 1/ln(b) ) + C.

6. Change of base rule

This rule is as follows:

  • Log b m = log a m/ loga b.

7. Base switch rule

The rule is as follows:

  • log b (a) = 1 / log a (b).

Properties of Logarithms

The properties are as follows:

No. Properties:
1. Log b is equal to 1.
2. Log 1 with base b is equal to 0.
3. Log 0 with base b is undefined.

Examples of Logarithmic Problems

The examples are as follows:

Example 1:

Calculate the possible answer of log 2 (64).


As 2^6 = 2x2x2x2x2x2x2 = 64, the exponent value is 6 and log 2 (64)= 6.

Example 2:

Find x such that log 2 x = 5.


The exponential form of this logarithmic function is 2^5 = x.
So, 2^5= 2x2x2x2 = 32.Hence, the value of x= 32.

Using Logarithms in the real life

The uses of logarithms in real life are as follows:

  1. Logarithms can help measure the earthquake intensity through an instrument known as Seismograph.
  2. Logarithms can measure the pH value of a substance.
  3. It can calculate the sound intensity in decibels.
  4. Logarithms help solve complex values.
  5. It helps find the half-life of radioactive substances.
  6. It has extensive use in radioactive decay.
  7. Logarithms can calculate the most challenging operations of division and multiplication.

Frequently Asked Questions

Below are the most frequent questions people ask about Fundamentals of Logarithms:

1. What are binary logs?

The binary logarithm is a base two logarithm used in computer science.

2. What is the definition of antilogarithm?

Anti-Logarithms is the reverse technique for calculating the logarithm of the same number. Consider this: if a is the logarithm of a number b with base x, then b is the antilog of a to the base x.

3. What is the value of the logarithm of 0?

The logarithm of zero is undefined because we can never get the value 0 by multiplying any value with anything else.

4. What is the value of the logarithm of 10?

The logarithm of ten equals one. As a result, the base ten logarithms of ten equals one.

5. What is the first law of the logarithm?

Adding log C and log D yields log CD, which is the logarithm of the product of C and D. It provides a way to add two logs. The formula is as follows:

  • log C + log D = log CD.

6. What is the second law of the logarithm?

According to the second law, the subtraction of log D from log C results in log C/D.

  • log C − log D = log C/D.

7. What is the third law of the logarithm?

The statement of the third law is as follows:

  • log C^n = n log C

8. Is a logarithm a real number?

A logarithm is not a real number, as you can never get zero by multiplying anything by anything else. You can never meet 0; all you can do is try with enormous and negative power.

9. Can the value of the log be negative?

The logarithm of a positive number might be negative or zero.

10. What are the names of the components of a logarithm?

The fractional part is the mantissa, and the integer part is the characteristic of the logarithm.

11. Can a log’s base be smaller than one?

Not at all. But, it must be different from one and greater than zero.

12. Is log irrational or rational?

In this brief note, we demonstrate that the decimal logarithm of every integer is irrational until it is a power of 10 and that the natural logarithm of every integer 2 is irrational. This brief essay demonstrates that most integer logarithms are irrational.

13. What exactly is the log of a negative number?

A negative number’s logarithm is calculated by multiplying its magnitude by -1.


Fundamentals of Logarithms:
  • A logarithm calculates how many times a base multiplies itself to get to another number.
  • Logaritimos is another word for logarithms.
  • In the 17th century, John Napier introduced the idea of logarithms.

Also Read

How to solve logarithmic equations
How to do logarithms
Simplifying logarithms