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### Number and algebra

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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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### Powerful Quadratics

### Discriminating

### Factorisable Quadratics

Links to the University of Cambridge website
Links to the NRICH website Home page

Nurturing young mathematicians: teacher webinars

30 April (Primary), 1 May (Secondary)

30 April (Primary), 1 May (Secondary)

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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?

This comes in two parts, with the first being less fiendish than the second. Itâ€™s great for practising both quadratics and laws of indices, and you can get a lot from making sure that you find all the solutions. For a real challenge (requiring a bit more knowledge), you could consider finding the complex solutions.

You're invited to decide whether statements about the number of solutions of a quadratic equation are always, sometimes or never true.

This will encourage you to think about whether all quadratics can be factorised and to develop a better understanding of the effect that changing the coefficients has on the factorised form.