Logarithms are used in various fields, such as mathematics, science, and computer science, to express exponential relationships. The base of a logarithm determines how the logarithm behaves.

Base 10 (Common Logarithm)

The common logarithm is the logarithm with base 10, often written as log(x) or log10(x).
Scaling Large Numbers: It's used for compressing large numbers, making them easier to interpret (e.g. in scientific notation).

Example:

log10(1000) = 3 because 103 = 1000.

Base e (Natural Logarithm)

The natural logarithm is the logarithm with base e >, where e ≈ 2.71828, often written as ln(x).
Mathematical Significance: e arises naturally in many areas of mathematics, especially with continuous growth or decay.
Calculus and Derivatives:The function ex has the special property that it is its own derivative, making it crucial for continuous processes.

Example:

ln(e) = 1 because e1 = e.

Common Applications:

• Exponential growth and decay (e.g. population growth, radioactive decay).
• Finance (e.g. compound interest, continuous compounding).

Base 2 (Binary Logarithm)

The binary logarithm is the logarithm with base 2, often written as log2(x).

Why Base 2?

• Computer Science: Base 2 is used because computers operate on binary, which involves powers of 2 (0 and 1).
• Doubling and Halving: Base 2 is used to measure data storage, algorithm complexity, and more in computing.

Example:

log2(16) = 4 because 24 = 16.

• Computer science (e.g., binary search, data structures).
• Data storage (e.g. bits, bytes).
• Cryptography (e.g. encryption algorithms).