The logarithm is the inverse operation of exponentiation. It answers the question: “To what exponent must a base be raised, to produce a certain number?”

Mathematically, the logarithm is written as:

$$\log_b(x) = y$$

This means that:

$$b^y = x$$

Where:

• ( b ) is the base (a positive number, other than 1),
• ( x ) is the argument (a positive number),
• ( y ) is the exponent or logarithm.

Example

$$\log_2(8) = 3$$

This is because:

$$2^3 = 8$$

Common Logarithms

1. Common logarithm: This uses base 10, written as ( \log_{10}(x) ), and is often abbreviated as just ( \log(x) ).

   Example:

   $$\log_{10}(100) = 2 \quad \text{because} \quad 10^2 = 100$$

2. Natural logarithm: This uses base ( e ) (approximately 2.718), written as ( \ln(x) ).

   Example:

   $$\ln(e^2) = 2$$

Key Properties of Logarithms

• Product Rule: ( \log_b(xy) = \log_b(x) + \log_b(y) )
• Quotient Rule: ( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) )
• Power Rule: ( \log_b(x^k) = k \log_b(x) )

Latex

\log_b(x) = y
b^y = x