Properties of Logarithms
The properties of logarithms are rules that govern how logarithms interact with multiplication, division, and exponents.
Definition
Logarithms follow special rules: multiplying inside the log becomes addition, dividing becomes subtraction, and exponents come out front. For any valid base $b$: the product rule $\log_b(xy) = \log_b(x) + \log_b(y)$, the quotient rule $\log_b(x/y) = \log_b(x) - \log_b(y)$, the power rule $\log_b(x^r) = r\log_b(x)$, plus $\log_b(b) = 1$ and $\log_b(1) = 0$. These properties follow from the isomorphism $\log_b: (\mathbb{R}^+, \times) \to (\mathbb{R}, +)$: a multiplicative identity becomes the additive identity, products become sums, and inverses become negatives, the algebraic content underlying all the logarithm rules.
Example
$\log(100 \times 10) = \log(100) + \log(10) = 2 + 1 = 3$, matching $\log(1000) = 3$ directly, an instance of the product rule turning multiplication into addition. Expanding $\log_2(8x^3/y) = \log_2(8) + 3\log_2(x) - \log_2(y) = 3 + 3\log_2(x) - \log_2(y)$, and condensing $2\ln(x) + \ln(5) - \ln(y) = \ln(5x^2/y)$, both apply these rules directly. The isomorphism $\log_b$ converting the multiplicative group $(\mathbb{R}^+, \times)$ to the additive group $(\mathbb{R}, +)$ makes the product rule $\log(xy) = \log(x) + \log(y)$ exactly the homomorphism property $\phi(xy) = \phi(x) + \phi(y)$.
Key Insight
These rules are why logarithms were historically used for computation, to multiply two large numbers, navigators and engineers just looked up their logs, added, and looked up the antilog of the sum, exactly how slide rules work. The properties reflect the corresponding exponent rules, $b^m \cdot b^n = b^{m+n}$ gives the product rule, $(b^m)^n = b^{mn}$ gives the power rule, since logarithms are exponents and obey the rules of exponents. Logarithms are in fact the unique continuous homomorphisms from $(\mathbb{R}^+, \times)$ to $(\mathbb{R}, +)$ up to scaling, an algebraic characterization more fundamental than the computational rules, explaining why logarithms appear universally wherever multiplicative structure needs to be studied additively.