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WAEC 2020 · Paper 1 · Q39

If log⁡5(125x3y3)\log_5\left(\dfrac{125x^3}{\sqrt[3]{y}}\right) is expressed in the form plog⁡5x+qlog⁡5y+kp\log_5 x + q\log_5 y + k, find the values of pp, qq and kk respectively.

Worked solution (try it first)
  1. Split the log of a product and quotient: log⁡5125+log⁡5x3−log⁡5y13\log_5 125 + \log_5 x^3 - \log_5 y^{\frac{1}{3}}.
  2. Bring the powers down: log⁡5125+3log⁡5x−13log⁡5y\log_5 125 + 3\log_5 x - \frac{1}{3}\log_5 y.
  3. 125=53125 = 5^3, so log⁡5125=3\log_5 125 = 3.
  4. The expression is 3log⁡5x−13log⁡5y+33\log_5 x - \frac{1}{3}\log_5 y + 3.
  5. So p=3p = 3, q=−13q = -\frac{1}{3}, k=3k = 3, option D.

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