Topic 3 of 9 · Form 3
Surds, indices, logarithms and exponentials
Simplify √50 = 5√2; rationalise denominators: 1/(√3 − 1) = (√3 + 1)/2.
Notes for this topic 1 file, in order
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Surds
Simplify √50 = 5√2; rationalise denominators: 1/(√3 − 1) = (√3 + 1)/2.
Indices
Laws of indices; solving equations by matching bases (e.g. 4ˣ = 8 → 2²ˣ = 2³ → x = 1.5).
Logarithms
logₐx = y ⇔ aʸ = x; laws: log(xy) = log x + log y; log(x/y) = log x − log y; log xⁿ = n log x; change of base logₐb = log b/log a; natural logarithms ln x and eˣ.
Worked example: Solve 5ˣ = 30.
x = ln 30/ln 5 = 2.11
Worked example: Solve log₂(x + 1) + log₂(x − 1) = 3.
log₂(x² − 1) = 3 → x² − 1 = 8 → x = 3 (x = −3 rejected)
Linear law (reducing to straight lines)
- y = axⁿ → lg y = n lg x + lg a (plot lg y against lg x).
- y = abˣ → lg y = x lg b + lg a.
Gradient and intercept give the constants.
Exponential graphs
y = eˣ and y = ln x are reflections in y = x; growth and decay models y = Ae^(kx).
Check yourself
- Simplify (√12 + √27).
- Express 2 lg 5 + lg 4 as a single number.
- Solve e^(2x) = 7.
Show answers
- 2√3 + 3√3 = 5√3
- lg 100 = 2
- x = ½ ln 7 = 0.973
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