(a) If ((y – 1)log_{10}4 = ylog_{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 = ylog_{10} 16)
((y – 1)log_{10} 4 = y log_{10} 4^{2})
((y – 1)log_{10} 4 = 2ylog_{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)