title: Why is the following sequence of functions ordered by asymptotic growth rates?
body: <p>Order the following expressions in increasing Θ-order. If two functions are of the same order of growth, you should state this fact.</p>
  
  <blockquote>
    <p>n log n, n<sup>−1</sup>, log n, n<sup>log n</sup>, 10n + n<sup>3/2</sup>, π<sup>n</sup>, 2<sup>n</sup>, 2<sup>log n</sup>, 2<sup>2<sup>log n</sup></sup>, log n!</p>
  </blockquote>
  
  <p>Can someone explain to me why the following answer is correct?</p>
  
  <blockquote>
    <p>n<sup>−1</sup> ≪ log n ≪ 2<sup>log n</sup> ≪ n log n = log n! ≪ 10n + n<sup>3/2</sup>≪ n<sup>log n</sup> ≪ 2<sup>n</sup> = 2<sup>2<sup>log n</sup></sup> ≪ π<sup>n</sup></p>
  </blockquote>