How do you prove log(n + 1) = log(n) + 1 for integers n >= 1???

Answer 1

Not true.

Assume the logs are base a. So log_a(x)= log base a of x.

Then: log_a(n)+1=log(n)+log_a(a) = log_a (n*a).

Now if log_a (n*a)= log_a(n+1), we have that na=n+1 so n(a-1)=1, which means n=1/(a-1). This only happens once for every base a. So the statement is not true for all n.

Answer 2

Where do you get that from? It's false
log(n+1) = log(n) + 1/n + O(1/n^2)

Answer 3

you can't. It's not so.

proof: let n=9

log(10)<>log(9) + 1

log(9) would have to be zero which it isn't