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TMUA 2019 · Paper 2 · Question 15 of 20

TMUA 2019 Paper 2 Question 15

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

TMUA 2019 · Paper 2Exponentials and logarithmsSimultaneous logarithm equations5 options
The numbers a, b and c are each greater than 1.

The following logarithms are all to the same base:

log(ab2c) = 7 log(a2bc2) = 11 log(a2b2c3) = 15

What is this base?

  1. Aa
  2. Bb
  3. Cc
  4. DIt is possible to determine the base, but the base is not a, b or c.
  5. EThere is insufficient information given to determine the base.
Show the answer and worked solution
answer · B
  1. Aa
  2. Bb
  3. Cc
  4. DIt is possible to determine the base, but the base is not a, b or c.
  5. EThere is insufficient information given to determine the base.
Put A = log a, B = log b, C = log c; the log laws turn the three equations into A + 2B + C = 7, 2A + B + 2C = 11 and 2A + 2B + 3C = 15. Subtracting the second from the third gives B + C = 4, and the first then reads A + B + (B + C) = 7, so A + B = 3. Feeding C = 4 − B into the second gives 2A − B = 3, and with A + B = 3 this gives A = 2, B = 1, C = 3. A logarithm equal to 1 means the number equals the base, so log b = 1 makes the base b.