Rules of Logarithms
This article presents the rules of logarithms using complete, line-by-line derivations. Every identity is built directly from its exponential origin, without shortcuts, matching the structure of formal handwritten algebra. 1. Definition We begin with fundamental exponent facts: a⁰ = 1 ⇒ logₐ(1) = 0 a¹ = a ⇒ logₐ(a) = 1 Say: aᵐ = p Then, by definition: logₐ(p) = m Raise both sides of aᵐ = p to the power 1/m (with m ≠ 0 ): p^(1/m) = a Therefore: logₚ(a) = 1/m Since m = logₐ(p) , we obtain: logₐ(p) = 1 / logₚ(a) 2. Product Rule — Full Derivation Say: aᵐ = p and aⁿ = q Multiply: aᵐ · aⁿ = p · q Using index addition: a^(m+n) = p · q Taking logarithms: logₐ(p · q) = m + n Substitute: logₐ(p · q) = logₐ(p) + logₐ(q) 3. Quotient Rule — Full Derivation Say: aᵐ = p and aⁿ = q Divide: aᵐ / aⁿ = p / q Index subtraction gives: a^(m−n) = p / q Taking logarithms: logₐ(p / q) = m − n So: log...