(a) If \((y - 1)\log_{10}4 = y\log_{10}16\), without using Mathematics tables or calculator, find the value of y. (b) When I walk from my house at 4km/h, I will get to my office 30mins later than when I walk at 5km/h. Calculate the distance between my house and office.
Explanation
(a) \((y - 1)\log_{10} 4 = y\log_{10} 16\) \((y - 1)\log_{10} 4 = y \log_{10} 4^{2}\) \((y - 1)\log_{10} 4 = 2y\log_{10} 4\) Equating both sides, we have \(y - 1 = 2y \implies -1 = 2y - y\) \(\therefore y = -1\) (b) Let the distance from my house to the office = c. At 4km/h, the time taken to get to the office from the house = \(\frac{c}{4} hr\) At 5km/h, the time taken to get to the office from the house = \(\frac{c}{5} hr\) \(\frac{c}{4} = \frac{c}{5} + \frac{30}{60}\) \(\frac{c}{4} - \frac{c}{5} = \frac{1}{2}\) \(\frac{c}{20} = \frac{1}{2} \implies c = 10km\)