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# Log Lattice

**This is an Underground Mathematics resource.**

*Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.*

*Visit the site at undergroundmathematics.org to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.*

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Age 16 to 18

Challenge Level

Can you arrange some of these logarithms to complete the grid below? The values of the logarithms need to increase along the rows and down the columns. Try to do this without using a calculator.

$$ \log_{3} 2 \quad \log_{4} 5 \quad \log_{2} 5 \quad \log_{3} 4 \quad \log_{3} 5 \quad \log_{5} 3 $$

$$ \log_{4} 2 \quad \log_{2} 4 \quad \log_{2} 3 \quad \log_{5} 2 \quad \log_{5} 4 \quad \log_{4} 3 $$

Think about

- Can you put pairs of logarithms from the list in order of size?
- Which logarithms are bigger than $1$ and which ones are smaller?