# Change in accessible states relating to change in energy

1. Oct 26, 2014

### leroyjenkens

1. The problem statement, all variables and given/known data
A certain system has 6 × 10^24 degrees of freedom. Its internal energy
increases by 1%. By what factor does the number of accessible states increase?

2. Relevant equations
$$\Omega = E^{N\nu/2}$$
$\nu$ is the degrees of freedom, and N is just 1, so we can ignore that. So the exponent is just 3x10^24
$\Omega$ is the number of accessible states.
3. The attempt at a solution

First I replaced E with 10. And then E increased by 1% would be 10.1.
What I was going to try to do was divide 10.1^(6x10^24) by 10^(6x10^24) to get my answer. But no calculator in the world can do that.
So I used a calculator that can do big numbers (but not quite that big). I found that as I increased the 0's in the exponent, the exponent of my answer increased by some seemingly random number. I tried 10.1^(300)/10^(300) and then 10.1^(3000)/10^(3000), and then kept adding zeroes like that to see what kind of pattern I got.
What I got was, as I got up to 10 zeroes, was an answer of 3x10^(129641213). If I take away a zero from the exponents in the fraction, then I just lose an exponent in the answer. So if I had 24 zeroes in my exponents in my fraction, then the answer would be roughly 3x10^(1.29x10^22).

But this can't be the way to do this problem. Anyone have an alternative idea on how to solve this? Thanks.

2. Oct 26, 2014

### BvU

Try taking logarithms on both sides: the desired ratio is $\Omega '/\Omega = \exp ...$