Log lattice
Can you figure out how each log fits into the lattice?
Age
16 to 18
Challenge level
Problem
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.
$$ \log_{3} 2 \quad \log_{4} 5 \quad \log_{2} 5 \quad \log_{3} 4 \quad \log_{3} 5 \quad \log_{5} 3 $$
$$ \log_{4} 2 \quad \log_{2} 4 \quad \log_{2} 3 \quad \log_{5} 2 \quad \log_{5} 4 \quad \log_{4} 3 $$
Image
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?
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.
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.
Getting Started
Which is the bigger:
Can you explain why this inequality is true?
- $\log_{2} 5$ or $\log_{2} 3$?
- $\log_{2} 5$ or $\log_{3} 5$?
Can you explain why this inequality is true?
- $\log_{5} 2 < 1 < \log_{2} 5$
Student Solutions
Thank you to everyone who submitted solutions to this problem.
Jessica form Tiffin Girls school has found a really nice way of organising all the given logarithms into a table.
Image
This table nicely captures all the information we can get about the relative sizes of the logarithms without calculating them, by making use of the fact that $\log_b{a}$ increases in size as you increase $a$ or decrease $b$.
It also makes it much easier to read off all the different ways you could complete the given grid, of which there are quite a few. Here is one way Jessica has given to complete the grid.
Image
You can read Jessica's full solution here .