Algorithm analysis: Choosing the best algorithm

I'm trying to solve a question about comparing 2 algorithms in terms of their worst case running times and finding the input size for which one algorithm would have a faster running time than the other.

The two algorithms are:
A₁ = 2n log₁₀ n
A₂ = 0.1n²

Basically, I am trying to solve the following inequality for n:
2n log₁₀ n < 0.1n²

Can anyone please point me in the right direction?
I have managed to get up to:
log₁₀ n < 0.05n ==> n < 10^0.05n

But I have no idea what to do from here (or perhaps I have gone about the wrong way trying to solve it).

Thank you for your help in advance!

Solution

Actually, you are trying to solve the inequality

point 1 n squared less than 2 times n log base 10 n

because the n squared algorithm is only going to be faster for a very short time, and then for any larger values of n, the n log n algorithm will be faster.

Ignore the case n <= 0, multiply by 10, and divide by n to produce:

n less than 20 log base 10 n

Then divide by 20 and exponentiate both sides with a base of 10:

10 to the n over 20 less than n

Use a numeric solver to find the zeros of 10 to the n over 20 minus n on the interval [1, 40] since clearly 40 is an upper bound (because 10 to the 40 over 20 equals 10 squared equals 10,040 ).

For instance, in Matlab:

>> fzero(@(x) 10^(x/20)- x, 20)

ans =

   29.3531

So for any n an integer up through 29, the n squared algorithm is faster, and for n > 29, the n log n algorithm wins.