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TMUA 2021 · Paper 2 · Question 14 of 20

TMUA 2021 Paper 2 Question 14

Exponentials and logarithms — Simultaneous equations with exponentials and logarithms. Try it first; the answer and a full worked solution are below.

TMUA 2021 · Paper 2Exponentials and logarithmsSimultaneous equations with exponentials and logarithms8 options
Consider the following simultaneous equations, where p is a real number: p 2x + log2 y = 2 2x + log2 y = 1 What is the complete range of p for which these simultaneous equations have a real solution (x, y)?
  1. Ap < 1
  2. Bp ≠ 1
  3. Cp > 1
  4. Dp < 1 or p > 2
  5. Ep ≠ 1 and p < 2
  6. Fp > 1 and p < 2
  7. Gp > 2
  8. HAll real values of p
Show the answer and worked solution
answer · C
  1. Ap < 1
  2. Bp ≠ 1
  3. Cp > 1
  4. Dp < 1 or p > 2
  5. Ep ≠ 1 and p < 2
  6. Fp > 1 and p < 2
  7. Gp > 2
  8. HAll real values of p
Treat 2x and log2 y as the two unknowns. Subtracting the second equation from the first eliminates the logarithm and gives (p−1)2x = 1, so 2x = 1p−1. The constraint is that 2x is strictly positive for every real x, which forces p − 1 > 0, that is p > 1. Nothing further is needed: once 2x is fixed, log2 y = 1 − 2x is some real number and y = 21−2x is a genuine positive value, so p > 1 is the complete range.