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TMUA 2019 · Paper 1 · Question 11 of 20

TMUA 2019 Paper 1 Question 11

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

TMUA 2019 · Paper 1Exponentials and logarithmsSimultaneous equations in logarithms8 options
Find the sum of the real values of x that satisfy the simultaneous equations log3(xy2) = 1 (log3 x)(log3 y) = −3
  1. A13
  2. B1
  3. C3
  4. D319
  5. E9127
  6. F913
  7. G27
  8. H2719
Show the answer and worked solution
answer · H
  1. A13
  2. B1
  3. C3
  4. D319
  5. E9127
  6. F913
  7. G27
  8. H2719
Put a = log3 x and b = log3 y, which turns the pair into a + 2b = 1 and ab = −3. Substituting a = 1 − 2b gives b − 2b2 = −3, so 2b2 − b − 3 = 0 and (2b−3)(b+1) = 0. If b = 32 then a = −2 and x = 3−2 = 19; if b = −1 then a = 3 and x = 27. The sum of the x values is 2719.