Formula to calculate profit over time

Question

I am wondering if there is a formula to calculate something like this:

Natascha has 10,000 usd and makes +1% profit every day. After how many days does she have 1,000,000 usd?

The problem is that she makes +1% profit over the amount of the previous day which is groing every day. So, the first day she makes 100 usd, the 2nd day she makes 101, etc.

The formula I al looking for is something like:

days = goal / (start amount + 1%)

So, what I got is:

days = 1,000,000 / (10,000 + 1%) which gives me the answer 99 days which is wrong according to my excel sheet.

Answer 1

Natascha has $10,000$ usd at the start. After one day, she has $10,000\times (1+0.01)$. After two days, she has $10,000\times (1+0.01)^2$, and after $n$ days, she has $10,000\times (1+0.01)^n$. Solving the equation $10000(1+0.01)^n=1000000$ gives the answer.

Answer 2

Thanks to uniquesolution:

$(1 + 0.01)^n$ has to be equal to $100$

So..

$log (1 + 0.01)^n = log(100)$

$n \times log (1 + 0.01) = log(100)$

$n = log(100)/log(1.01)$

$n = 462,8$ days