Summing to one
This problem is a nice introduction that will give you a feeling for how logs work and what that button on your calculator might be doing.
Age
16 to 18
Challenge level
Problem
$$\log_3 3=1$$
$$\log_9 3+ \log_9 3 =1$$
$$\log_{27} 3 + \dots = 1$$
How many $\log_{81} 3$ do you need to add together to make one?
Can we choose integers $x$ and $y$ so that:
$$\log_6 x +\log_6 y =1?$$
How many different ways are there to do this?
How about using $\log_{12}$?
How about using $\log_{24}$?
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.
Student Solutions
Well done to Tiffany from Cardiff Sixth Form College, Jack from Gatesehead, Raj, Leon from Hitchin Boys School, Deepak from KV Dogra Lines, Niharika from Rugy School, Emily, Christian from Thoren Business School, and Julian from the British School Manila who all submitted correct solutions to this problem.
Here is Deepak's solution.
This is an Underground Mathematics resource.