Logarithm Formula, Laws and Properties Explained with Examples

Logarithms are the inverse of exponentiation. They were developed to simplify complex calculations involving multiplication, division, powers, and roots.

Even today, logarithms are extremely important in mathematics, physics, chemistry, biology, and engineering. They help in solving exponential equations, analysing growth and decay, and working with large numerical ranges.

Meaning of Logarithm

If,

aˣ = b

then,

logₐ b = x

This means a logarithm tells us the power to which a base must be raised to obtain a number.

Example

2³ = 8
⇒ log₂ 8 = 3

Here:

Base = 2
Number = 8
Logarithm = 3

Conditions for Logarithms

For a logarithm to exist:

Base a > 0
Base a ≠ 1
Number b > 0

Logarithm of zero or negative numbers is not defined (in real numbers).

Common Types of Logarithms

1. Common Logarithm

Base = 10
Written as log x

Example: log 100 = 2

2. Natural Logarithm

Base = e ≈ 2.718
Written as ln x

Example: ln e = 1

Widely used in calculus, growth and decay problems.

Laws of Logarithms

These laws are very important for simplification.

1. Product Law

logₐ (MN) = logₐ M + logₐ N

2. Quotient Law

logₐ (M/N) = logₐ M − logₐ N

3. Power Law

logₐ (Mᵏ) = k logₐ M

4. Change of Base Formula

logₐ b = (log_c b) / (log_c a)

Used to evaluate logs using base 10 or base e.

Characteristic and Mantissa

A common logarithm has two parts:

Characteristic → Integer part
Mantissa → Decimal part

Example

log 500 = 2.69897

Characteristic = 2
Mantissa = 0.69897

For numbers less than 1, the characteristic is negative.

Logarithmic Scale

Logarithmic scales are used to handle very large or very small values.

Examples:

pH scale
Richter scale
Decibel scale

Each step represents multiplication, not addition.

Solving Logarithmic Equations

Example 1

log x = 2
⇒ x = 10² = 100

Example 2

log₂ x = 5
⇒ x = 2⁵ = 32

Example 3

log x + log (x − 3) = 1

Using product law:

log [x(x − 3)] = 1

⇒ x(x − 3) = 10

⇒ x² − 3x − 10 = 0

⇒ (x − 5)(x + 2) = 0

Possible values: x = 5, −2

But:

x − 3 > 0 ⇒ x > 3

Final answer: x = 5

Exponential and Logarithmic Relationship

Logarithmic and exponential functions are inverse functions.

If: y = aˣ

Then: x = logₐ y

Graphical Insight

Exponential function → increases rapidly
Logarithmic function → increases slowly

They are reflections across the line y = x

Logarithmic Functions

General form: y = logₐ x

Properties

Domain: x > 0
Range: All real numbers
Passes through (1, 0)
Vertical asymptote: x = 0
Increasing if a > 1
Decreasing if 0 < a < 1

Important Log Identities

logₐ 1 = 0
logₐ a = 1
a^(logₐ x) = x
logₐ (aˣ) = x

Limitations of Logarithms

log of negative numbers is undefined (real system)
log 0 is undefined
Cannot take the log of quantities with units directly

Conclusion

Logarithms convert complex exponential relationships into simpler forms. They are powerful tools for solving equations, analysing growth, and simplifying calculations.

Understanding logarithms is essential not just for exams, but for deeper concepts in physics, chemistry, and higher mathematics.