Advanced Algebra & Functions

Functions, exponentials, logarithms and complex equations

Explanation

Exponentials & Logarithms

  • Exponential growth: y = aᵇˣ where b > 1; decay: 0 < b < 1
  • log_b(x) = y means bʸ = x
  • ln = natural log (base e)
  • Laws: log(ab) = log a + log b; log(aⁿ) = n·log a; log(a/b) = log a − log b

Rational Expressions

  • Simplify by factoring numerator and denominator, then cancel common factors.
  • Adding/subtracting: find LCD, rewrite with common denominator.
  • Restrictions: denominator ≠ 0 — always state excluded values.

Absolute Value & Radicals

  • |x| = a → x = a or x = −a (two solutions)
  • |x| < a → −a < x < a (interval)
  • √(a²) = |a|, not a (important for even-degree roots)
  • Rationalise the denominator: multiply by √conjugate
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