Taking it further

We have extended one possible solution grid. Can you use other logarithms to complete the four blank cells of this grid or one based on your own solution grid?

Increasing size \(\longrightarrow\)
\(\log_{\square} \square\)
\(\log_{4} 2\) \(\log_4 3\) \(\log_{4} 5\)
\(\log_{\square} \square\) \(\log_{5} 3\) \(1\) \(\log_{3} 5\) \(\log_{\square} \square\)
\(\log_{5} 4\) \(\log_3 4\) \(\log_{2} 4\)
\(\log_{\square} \square\)

Some things to think about:

  • Try to make your extra logarithms as general as possible. For instance if you want \(\log_{a} 3\) to be less than \(\log_{5} 3\), what conditions must \(a\) satisfy?
  • How could you include fractions or irrational numbers in your logarithms?