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# Introduction to Logarithms – Explanation & Examples

Before getting into the topic of logarithms, it is important we briefly discuss exponents and powers.

The exponent of a number is the frequency or number of times a number is multiplied by itself. An expression that represents repeated multiplication of the same factor is called a power.

For example, the number 16 can be expressed in exponential form as; 2^{4}. In this case, the numbers 2 and 4 are the base and exponent, respectively.

## What is a Logarithm?

On the other hand, the **logarithm of a number is the power or index to which a given base must be raised to obtain the number.**

The concept of logarithm was introduced in the 17^{th} century by a Scottish mathematician named **John Napier**.

It was introduced to mechanical machinery in the 19^{th} century and to computers in the 20^{th} century. **The natural logarithm** is one of the useful functions in mathematics and has many applications.

Consider three numbers a, x and n, which are related as follows;

a^{x} = M; where a > 0 < M and a ≠ 1

The number x is the logarithm of the number n to the base ‘a’. Therefore, a^{x} = n can be expressed in logarithmic form as.

log _{a }M = x, Here, M is the argument or the number; x is the exponent while ‘a’ is the base.

**For example:**

16 = 2 ^{4 }⟹ log _{2} 16 = 4

9 = 3^{2} ⟹ log _{3} 9 = 2

625 = 5^{4} ⟹ log _{5} 625 = 4

7^{0} = 1 ⟹ log _{7} 1 = 0

3^{– 4} = 1/3^{4} = 1/81 ⟹ log _{3} 1/81 = -4

### The common logarithms

All the logarithms with base 10 are called **common logarithms**. Mathematically, the common log of a number x is written as:

log _{10 }x = log x

### The natural logarithms

A** natural logarithm** is a special form of logarithms in which the base is mathematical constant e, where e is an irrational number and equal to 2.7182818…. Mathematically, the natural log of a number x is written as:

log _{e }x = ln x

where the natural log or *ln* is the inverse of *e*.

The natural exponential function is given as:

e ^{x}

### The negative logarithms

We know that logarithms are not defined for negative values.

*Then what do we mean by the negative logarithms? *

It means that the logarithm of the set of such numbers gives a negative result. All the numbers that lie between 0 and 1 have negative logarithms.

### Basic Laws of Logarithms

There are four basic rules of logarithms. These are:

**Product rule.**

The product of two logarithms with a common base is equal to the sum of individual logarithms.

⟹ log _{b} (m n) = log _{b} m + log _{b} n.

**Division rule**

The division rule of logarithms states that the quotient of two logarithmic values with the same bases is equal to each logarithm’s difference.

⟹ log _{b} (m/n) = log _{b} m – log _{b }n

**The exponential rule of logarithms**

This rule states that the logarithm of a number with a rational exponent is equal to the product of the exponent and its logarithm.

⟹ log _{b} (m ^{n}) = n log _{b} ^{m}

**Change of Base**

⟹ log_{ b }a = log _{x }a ⋅ log _{b }x

⟹ log _{b }a = log _{x }a / log _{x }b

NOTE: The logarithm of a number is always stated together with its base. If the base is not given, it is assumed to be 10.

For example, log 100 = 2.

### Real-life application of logarithms

Logarithms very useful in the field of science, technology, and mathematics.

*Here are a few examples of real-life applications of logarithms.*

- Electronic calculators have logarithms to make our calculations much easier.
- Logarithms are used in surveys and celestial navigation.
- Logarithms can be used to calculate the level of noise in decibels.
- Ratio active decay, acidity [PH] of a substance and Richter scale are all measured in logarithmic form.

Let’s solve a few problems involving logarithms.

*Example 1*

Solve for x in log _{2} (64) = x

__Solution__

Here, 2 is the base, x is the exponent and 64 is the number.

Let 2^{x }= 64

Express 64 to the base of 2.

2^{x }= 2 × 2 × 2 × 2 × 2 × 2 = 2^{6}

x = 6, therefore, log _{2} 64 = 6.

*Example 2*

Find x in log_{10} 100 = x

__Solution__

100 = number

10 = base

x = exponent

Therefore, 10 ^{x} = 100

Hence x = 2

But 100 = 10 * 10 = 10^{2}

*Example 3*

Solve for k given, log_{3} x = log_{3} 4 + log_{3} 7

__Solution__

By applying the product rule log _{b} (m n) = log _{b} m + log _{b} n we get;

⟹ log_{3} 4 + log_{3} 7= log _{3} (4 * 7) = log _{3 }(28).

Hence, x = 28.

*Example 4*

Solve for y given, log _{2} x = 5

__Solution__

Here, 2 = base

x = number

5 = exponent

⟹ 2^{5} = x

⟹ 2* 2 * 2 * 2 * 2 = 32

Thus, x = 32

*Example 5*

Solve for log _{10 }105 given that, log _{10} 2 = 0.30103, log _{10} 3 = 0.47712 and log _{10} 7 = 0.84510

__Solution__

log_{10} 105 = log_{10} (7 x 5 x 3)

Apply the product rule of logarithms

= log_{10} 7 + log_{10} 5 + log_{10} 3

= log_{10} 7 + log_{10} 10/2 + log_{10} 3

= log_{10} 7 + log_{10} 10 – log_{10} 2 + log_{10} 3

= 0.845l0 + 1 – 0.30103 + 0.47712

= 2.02119.