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TMUA 2022 · Paper 2 · Question 15 of 20

TMUA 2022 Paper 2 Question 15

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

TMUA 2022 · Paper 2Exponentials and logarithmsChained logarithm equations8 options
The real numbers x, y and z are all greater than 1, and satisfy the equations logx y = z  and  logy z = x Which one of the following equations for logz x must be true?
  1. Alogz x = y
  2. Blogz x = 1y
  3. Clogz x = xy
  4. Dlogz x = 1xy
  5. Elogz x = xz
  6. Flogz x = 1xz
  7. Glogz x = yz
  8. Hlogz x = 1yz
Show the answer and worked solution
answer · F
  1. Alogz x = y
  2. Blogz x = 1y
  3. Clogz x = xy
  4. Dlogz x = 1xy
  5. Elogz x = xz
  6. Flogz x = 1xz
  7. Glogz x = yz
  8. Hlogz x = 1yz
The first equation says y = xz. Substituting into the second, z = yx = (xz)x = xzx. Taking logarithms to base x gives logx z = zx, and since logz x is its reciprocal, logz x = 1xz.