Can you, without using a calculator, arrange these logarithms into order of increasing value?

\[\log_{4} 5 \quad \log_{2} 5 \quad \log_{3} 4 \quad \log_{3} 3 \quad \log_{3} 5 \quad \log_{2} 4 \quad \log_{2} 3\]

  • Do you know exact values for any of them?
  • Can you find upper and lower bounds for them by comparing with known values?
  • Can you compare some of them by thinking about increasing and decreasing functions?
  • How might changing the base of logarithms help?

If you have completed Logarithm lattice, you will have put some of these logarithms in order already, but here we explore some different methods.

You might find it helpful to have the logarithms written or printed on cards that you can rearrange as you sort them.