1) Find a/b when \(2\log{(a -2b)} = \log{a} + \log{b}.\)
2) For each positive integer p, let b(p) denote the unique positive integer k such that \(|k-\sqrt{p}|<\frac{1}{2}\). For example, b(6)=2 and b(23)=5. Find \(S=\sum_{p=1}^{2007} b(p)\).
3) William Sydney Porter tried to perform the calculation \(\frac{-3+4i}{1+2i}\). However, he accidentally missed the minus sign, finding \(\frac{3+4i}{1+2i}=\frac{11}{5}-\frac{2}{5}i\). What answer should he have obtained?
+128475 1.
2log ( a - 2b) = log a + log b
log(a - 2b)^2 = ;og (ab) which implies that
(a - 2b)^2 = ab
a^2 - 4ab + 4b^2 = ab
a^2 - 5ab + 4b^2 = 0 factor
(a - 4b) ( a - b) = 0
So...either a - 4b = 0 ⇒ a = 4b ⇒ a / b = 4 (1)
or
a - b = 0 ⇒ a = b
If a is positive.....the original log on the left hand side is undefined
If a is negative......log a is undefined..so...
a / b = 4
