🔍 What is a Logarithm?

A logarithm (or “log” for short) is the inverse operation of exponentiation. In simple terms, a logarithm tells you what power a specific base must be raised to in order to get a certain number. For example:

x = by  →  y = logb(x)

Here, b is the base, x is the result, and y is the exponent (or the logarithm of x with base b).

When no base is specified, it usually means base 10, also called the common logarithm. If the base is e (Euler’s number, approximately 2.718), we write it as ln(x), the natural logarithm. Another frequently used base is 2, known as the binary logarithm, used in computer science.

⚙️ Where Are Different Logarithm Bases Used?

• Base 10 (log₁₀) – Common in science and engineering
• Base e (ln) – Used widely in mathematics, physics, and calculus
• Base 2 (log₂) – Essential in computing and information theory

📘 Basic Rules of Logarithms

1. Product Rule

When multiplying two numbers inside a log, you can rewrite it as the sum of their individual logs:

logb(x × y) = logb(x) + logb(y)

Example:
log(1 × 10) = log(1) + log(10) = 0 + 1 = 1

2. Quotient Rule

When dividing numbers inside a log, subtract the logs:

logb(x / y) = logb(x) - logb(y)

Example:
log(10 / 2) = log(10) - log(2) = 1 - 0.301 = 0.699

3. Power Rule

If the log argument has an exponent, pull the exponent in front:

logb(xy) = y × logb(x)

Example:
log(2⁶) = 6 × log(2) ≈ 6 × 0.301 = 1.806

4. Change of Base Rule

You can convert a logarithm from one base to another:

logb(x) = logk(x) / logk(b)

Example:
log₁₀(x) = log₂(x) / log₂(10)

5. Base-Argument Switch Rule

Flip the base and the argument using:

logb(c) = 1 / logc(b)

Example:
log₅(2) = 1 / log₂(5)

🧠 Other Logarithm Properties to Know

• log_b(1) = 0
• log_b(b) = 1
• log_b(0) = undefined
• lim_x→0 log_b(x) = −∞
• ln(e^x) = x