Logarithms are governed by a set of powerful rules that simplify complex calculations involving exponents, products, quotients, and powers. These formulas are derived directly from the laws of exponents.

Basic Logarithm Identities

These are the simplest formulas, derived from the definition of a logarithm $x = \log_a N \iff a^x = N$.

1. Logarithm of 1:
   $$\log_a 1 = 0 \quad (\text{Since } a^0 = 1)$$
2. Logarithm of the Base:
   $$\log_a a = 1 \quad (\text{Since } a^1 = a)$$

Key Properties (Laws of Logarithms)

These rules allow you to expand or compress logarithmic expressions.

3. Product Rule: The logarithm of a product is the sum of the logarithms.
   $$\log_a (mn) = \log_a m + \log_a n$$

   Proof Sketch: If $m=a^x$ and $n=a^y$, then $mn = a^x a^y = a^{x+y}$. Converting to logarithmic form gives $\log_a (mn) = x+y = \log_a m + \log_a n$.

4. Quotient Rule: The logarithm of a quotient is the difference of the logarithms.
   $$\log_a \left(\frac{m}{n}\right) = \log_a m – \log_a n$$

   Note: $\log_a \left(\frac{m}{n}\right) = \log_a m + \log_a n – \log_a n – \log_a n$ is an incorrect expansion based on the Product Rule shown in the provided image. The correct formula is $\log_a \left(\frac{m}{n}\right) = \log_a m – \log_a n$.

5. Power Rule (Argument Exponent): The logarithm of a number raised to an exponent is the exponent times the logarithm of the number.
   $$\log_a m^n = n \log_a m$$

Advanced Logarithm Formulas

These formulas relate logarithms with different bases or simplify complex exponents.

6. Power Rule (Base Exponent): The exponent on the base moves to the front as its reciprocal.
   $$\log_{a^n} b = \frac{1}{n} \log_a b$$
7. Combination Power Rule:
   $$\log_{a^n} b^m = \frac{m}{n} \log_a b$$
8. Inverse Logarithm Rule: The reciprocal of $\log_b a$ is $\log_a b$.
   $$\log_b a = \frac{1}{\log_a b}$$
9. Change of Base Formula (Base $a$ to Base $b$):
   $$\log_a b = \frac{\log_c b}{\log_c a}$$

   Note: This can be expressed using any new base $c$, but is commonly written using base 10 or base $e$ (natural log, $\ln$).

   $$\log_a b = \frac{\log b}{\log a} = \frac{\ln b}{\ln a}$$
10. Chain Rule (Multiplying Logarithms):
    $$\log_a b \cdot \log_b c = \log_a c$$

    Note: This rule can be extended to multiple terms, such as $\log_a b \cdot \log_b c \cdot \log_c d = \log_a d$.

11. Fundamental Logarithmic Identity: An exponential term whose exponent is a logarithm with the same base simplifies to the argument of the logarithm.
    $$a^{\log_a N} = N$$