title: log(n) vs log(k) in runtime of an algorithm with k < n
body: <p>I am having trouble understanding the difference between log(k) and log(n) in complexity analysis.</p>
  
  <p>I have an array of size n. I have another number k &lt; n that is an input of the algorithm (so it's not a constant known ahead of time). What are some examples of algorithms that would have log(n) vs those that would have log(k) in their complexity? I can only think of algorithms that have log(n) in their complexity.</p>
  
  <p>For example, mergesort has log(n) complexity in its runtime analysis (O(nlogn)).</p>