Properties of Logarithms
Functions & Advanced AlgebraThe 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.
Example
$\log(100 \times 10) = \log(100) + \log(10) = 2 + 1 = 3$. Check: $\log(1000) = 3$. The product rule turns multiplication into addition.
Key Insight
These rules are why logarithms were historically used for computation: instead of multiplying large numbers, navigators and engineers added their logarithms. Slide rules work on this principle.
Definition
Key logarithm properties (for any valid base $b$): Product: $\log_b(xy) = \log_b(x) + \log_b(y)$. Quotient: $\log_b(x/y) = \log_b(x) - \log_b(y)$. Power: $\log_b(x^r) = r\log_b(x)$. Identity: $\log_b(b) = 1$. Zero: $\log_b(1) = 0$.
Example
Expand $\log_2(8x^3/y)$: $= \log_2(8) + \log_2(x^3) - \log_2(y) = 3 + 3\log_2(x) - \log_2(y)$. Condense $2\ln(x) + \ln(5) - \ln(y)$: $= \ln(x^2) + \ln(5) - \ln(y) = \ln(5x^2/y)$.
Key Insight
The properties reflect the corresponding exponent rules: $b^m \cdot b^n = b^{m+n}$ corresponds to the product rule; $(b^m)^n = b^{mn}$ corresponds to the power rule. Logarithms are exponents, so they obey the rules of exponents.
Definition
The logarithm 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. This group isomorphism is the algebraic content of all the logarithm rules.
Example
The isomorphism $\log_b$ converts the multiplicative group $(\mathbb{R}^+, \times)$ to the additive group $(\mathbb{R}, +)$. The product rule $\log(xy) = \log(x) + \log(y)$ is exactly the homomorphism property $\phi(xy) = \phi(x) + \phi(y)$.
Key Insight
Logarithms are the unique continuous homomorphisms from $(\mathbb{R}^+, \times)$ to $(\mathbb{R}, +)$ up to scaling. This algebraic characterization is more fundamental than the computational rules and explains why logarithms appear universally wherever multiplicative structure needs to be studied additively.