Indices and logarithms — A/L Combined Mathematics

What must I remember about Indices and logarithms?

• Laws of Indices: a^m * aⁿ = a^m+n; a^m / aⁿ = a^m-n; (a^m)ⁿ = a^mn; (ab)ⁿ = aⁿ * bⁿ; a⁰ = 1 (a ≠ 0); a^-n = 1 / aⁿ; a^p/q = q√(a^p).
• Logarithm definition: If a^x = y with a > 0, a ≠ 1, and y > 0, then x = log_a(y); exponential and logarithmic functions are mutual inverses.
• Laws of Logarithms: log_a(xy) = log_a(x) + log_a(y); log_a(x/y) = log_a(x) - log_a(y); log_a(x^k) = k * log_a(x); log_a(a) = 1; log_a(1) = 0.
• Change of Base formula: log_b(x) = log_a(x) / log_a(b); directly implies log_b(a) = 1 / log_a(b) and log_b(x) * log_x(b) = 1.
• Solving exponential equations: Take natural logs (ln) or common logs (log10) on both sides: a^x = b^y => x * ln(a) = y * ln(b).

Test yourself (1 minute)

1. Logarithmic domain constraints

   For log_a(f(x)) to be defined, strictly enforce f(x) > 0, a > 0, and a ≠ 1; always verify potential solutions against original domain to eliminate extraneous roots.

2. Inequality base reversal

   If a > 1, then log_a(x) < log_a(y) <=> x < y; but if 0 < a < 1, the inequality direction REVERSES: log_a(x) < log_a(y) <=> x > y.

3. Natural log and Euler's e

   ln(x) = log_e(x) where e ≈ 2.71828; e^{ln x} = x and ln(e^x) = x for all valid x.

Practise Indices and logarithms questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 6

What should I study first?

What comes next?

Next in the syllabus

Where does Indices and logarithms fit in the syllabus?

Year 1 · Term 1