(a) Without using mathematical tables or calculator, simplify: (frac{log_28 + log_216 – 4 log_22}{log_416})
(b) If 1342(_{five}) – 241(_{five}) = x(_{ten}), find the value of x.
Explanation
(a) (frac{log_28 + log_216 – 4log_22}{log_416}) = (frac{log_22^3 + log_2 2^4 – 4log_22}{log_44^2})
= (frac{3 log_2 2 + 4log_22 – 4 log_2^2}{2 log 4^4})
But log(_2) 2 = (log_44 = 1)
Then, (frac{log_28+log_216 – 4 log_2^2}{log_416}) = (frac{3 + 4 – 4}{2}) = 1(frac{1}{2})
(b) Subtract 241(_{five}) to get 1101(_{five}). Thereafter, converting to base ten to find the value of x = (5(^3) x 1) + (5(^2) x 1 + (5(^1) x 0) + (5(^o) x 1) = 125 + 25 + 0 + 1 = 151
(b) They subtracted 241(_{five}) from 1342(_{five}) to get 1101(_{five}). Thereafter, converting to base ten to find the value of x = (5(^3) x 1) + (5(^2) x 1) + (5(^1) x 0) + (5°x 1) = 125 + 25 + + 1 = 151.