Indices and Logarithms
Indices and logarithms test fluency with the laws of indices, the equivalence between exponential and logarithmic form, and solving equations that mix the two. SPM also expects change of base and spotting when a substitution such as y = 2ˣ turns an equation into a quadratic.
Laws you must know
aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ; a^(m/n) = ⁿ√(aᵐ). The matching log laws are logₐ(xy) = logₐx + logₐy; logₐ(x/y) = logₐx − logₐy; logₐ(xⁿ) = n logₐx; logₐa = 1; logₐ1 = 0; and change of base logₐb = log꜀b / log꜀a.
Worked example
Solve log₂(x+3) + log₂(x−1) = 5. Combine the left side: log₂[(x+3)(x−1)] = 5, so (x+3)(x−1) = 2⁵ = 32. Then x² + 2x − 35 = 0, i.e. (x+7)(x−5) = 0, giving x = 5 or x = −7. Reject x = −7 because x+3 and x−1 must both be positive. The answer is x = 5.
Traps
Not checking that the arguments of the logarithms are positive; writing log(a+b) = log a + log b; forgetting logₐ1 = 0; combining terms with different bases without converting first.