log change formula
Can someone please help me with these optimisation questions finding it very difficult (ps have given what i think is valid answer to q1 but cant do q2 at all.)
note: 2^6 measn 2 to the power of 6,
note: theta is a capital theta
note: log c (a) means log to the base c of a
1) Let a, b, c be positive real numbers. Show that loga(b) = logc(b)/ logc(a).
( i think i have the answer to this can someone pls confirm this is a thoroguh enough proof
let y= log a (b)
then a^y =b
so taking logarithms to the base c gives
log c (a^y)= log c (b)
ylog c(a) = log c (b)
so: y = log c (b)/log c (a)
2) Let a, b be real numbers, with a>0 and b > 0. Show (n + a)^b = theta(n^b). Show that (log n)^b = o(n^a). (For concreteness let’s take ‘log’ to mean ‘log2’.