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TMUA 2019 · Paper 1 · Question 15 of 20

TMUA 2019 Paper 1 Question 15

Exponentials and logarithms — Exponential equation · quadratic in 3x. Try it first; the answer and a full worked solution are below.

TMUA 2019 · Paper 1Exponentials and logarithmsExponential equation · quadratic in 3x6 options
Find the real non-zero solution to the equation 2(9x)8(3x) = 14
  1. Alog3 2
  2. B2log3 2
  3. C1
  4. D2
  5. Elog2 3
  6. F2log2 3
Show the answer and worked solution
answer · A
  1. Alog3 2
  2. B2log3 2
  3. C1
  4. D2
  5. Elog2 3
  6. F2log2 3
Write everything as a power of 2: 8(3x) = 23 ⋅ 3x, so the left side is 29x − 3⋅ 3x and the equation becomes 9x − 3⋅ 3x = −2. Now put u = 3x, noting 9x = u2, to get u2 − 3u + 2 = 0, so u = 1 or u = 2. The root u = 1 gives x = 0, which is excluded, so 3x = 2 and x = log3 2.