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TMUA 2023 · Paper 2 · Question 6 of 20

TMUA 2023 Paper 2 Question 6

Exponentials and logarithms — Equations with the same number of solutions. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 2Exponentials and logarithmsEquations with the same number of solutions8 options
Consider the following equation where a is a real number and a > 1: (×)   ax = x Which of the following equations must have the same number of real solutions as (×)?

I   loga x = x
II  a2x = x2
III a2x = 2x

  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Show the answer and worked solution
answer · F
  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
For I, the graphs y = ax and y = loga x are reflections of each other in y = x, and a point where either meets y = x is fixed by that reflection — so the two equations have the same solutions. For III, putting u = 2x turns it into au = u, the same equation in a new letter, so the count is unchanged. For II, taking square roots gives ax = |x|: the positive branch reproduces (×), but the negative branch ax = −x always contributes exactly one extra solution. So II has one more.