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TMUA 2017 · Paper 1 · Question 14 of 20

TMUA 2017 Paper 1 Question 14

Exponentials and logarithms — Simultaneous exponential equations · difference of squares. Try it first; the answer and a full worked solution are below.

TMUA 2017 · Paper 1Exponentials and logarithmsSimultaneous exponential equations · difference of squares6 optionshard
The solution of the simultaneous equations 2x + 3 × 2y = 3 22x − 9 × 22y = 6 is x = p, y = q.

Find the value of p − q.

  1. A512
  2. B73
  3. Clog2 512
  4. Dlog2 73
  5. Elog2 9
  6. Flog2 15
Show the answer and worked solution
answer · F
  1. A512
  2. B73
  3. Clog2 512
  4. Dlog2 73
  5. Elog2 9
  6. Flog2 15
Put u = 2x and v = 2y. The equations become u + 3v = 3 and u2 − 9v2 = 6. The second is a difference of two squares: (u − 3v)(u + 3v) = 6, and since u + 3v = 3 this gives u − 3v = 2. Adding and subtracting, u = 52 and 3v = 12, so v = 16. Then p − q = log2 u − log2 v = log2uv = log2 15.