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CMA Foundation · Fundamentals of Business Mathematics and Statistics · Indices and Logarithms

If log base 2 of x + log base 2 of (x - 2) = 3, with x > 2, what is the value of x?

The value of x is 4. Combining the logarithms gives x times (x minus 2) equals 2 cubed, which is 8. The resulting quadratic factorises to (x minus 4)(x plus 2) equals zero, and only x equals 4 keeps both logarithm arguments positive.

  1. A2
  2. B4Correct
  3. C-2
  4. D6

Explanation

Using the product law, log2 [x(x - 2)] = 3, so x(x - 2) = 8 giving x^2 - 2x - 8 = 0. Factorising, (x - 4)(x + 2) = 0, so x = 4 or -2. The value -2 is rejected as logarithms need positive arguments (x > 2). Check: log2 4 + log2 2 = 2 + 1 = 3. The option 6 gives 6 x 4 = 24, not 8.

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