Definition of Logarithm

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

\[ \log_2(8) = 3 \]\[ 2^3 = 8 \]

Common Logarithms

Common logarithm: This uses base 10, written as \( \log_{10}(x) \), and is often abbreviated as just \( \log(x) \).
\[ \log_{10}(100) = 2 \quad \text{because} \quad 10^2 = 100 \]
Natural logarithm: This uses base \( e \) (approximately 2.718), written as \( \ln(x) \).
\[ \ln(e^2) = 2 \]

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