Logarithms and Exponential Functions

Common Logarithm: Base 10, written as log ⁡ ( x ) \log(x) lo g ( x ) .
Natural Logarithm: Base e e e (approx 2.718), written as ln ⁡ ( x ) \ln(x) ln ( x ) .

A logarithm answers the question: "To what power must a base be raised to produce a given number?" It is the inverse operation of exponentiation.

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

For $b > 0, b \neq 1$ , and positive real numbers $M, N$ :

Product Rule: $\log_b(MN) = \log_b(M) + \log_b(N)$
Quotient Rule: $\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$
Power Rule: $\log_b(M^p) = p \log_b(M)$
Change of Base: $\log_b(M) = \frac{\log_c(M)}{\log_c(b)}$ (usually $c=10$ or $e$ )
Identity: $\log_b(b) = 1$ and $\log_b(1) = 0$

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