Logarithm
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Table of Content:
Logarithm Formulas
Definition
A logarithm is the power to which a number must be raised to obtain another number. If \( b^y = x \), then \( \log_b(x) = y \).
\[
\log_b(x) = y \iff b^y = x
\]
Common Logarithm Properties
Product Rule
\[ \log_b(x \cdot y) = \log_b(x) + \log_b(y) \]Quotient Rule
\[ \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \]Power Rule
\[ \log_b(x^y) = y \cdot \log_b(x) \]Change of Base Formula
\[ \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \]Special Logarithms
Common Logarithm (Base 10)
\[ \log_{10}(x) = \log(x) \]Natural Logarithm (Base \( e \))
\[ \log_e(x) = \ln(x) \]Logarithm of 1
\[
\log_b(1) = 0 \quad \text{for any base } b
\]
Logarithm of the Base
\[
\log_b(b) = 1
\]
Logarithm of a Reciprocal
\[
\log_b\left(\frac{1}{x}\right) = -\log_b(x)
\]