To log or not to log?
Which of these logarithmic challenges can you solve?
Age
16 to 18
Challenge level
Problem
Some equations involving powers or indices can be solved using logarithms... but not all.
Think about how you could go about solving the following equations. Sort them according to the tools or methods you would use.
| $3^x=81$ | $x^5=50$ | $3^x=43$ | $5^{2x}-5^x-6=0$ |
| $5^x+4^x=8$ | $5^x+2\times5^{1-x}=7$ | $3^{2x}-3=24$ | $2^{2x}-9\times2^x+8=0$ |
| $\sqrt{2x-3}=5$ | $5^x-x^5=3$ | $16^{\frac{3}{x}}=8$ | $\big(\frac{13}{16}\big)^{3x}=\frac{3}{4}$ |
You might find it helpful to have the equations printed on cards that you can rearrange as you sort them. They are available here: cards.pdf
Can you write some other equations to go in each of your sorting categories?
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
You could use the following headings to categorise your equations
- Solve using indices
- Solve using logarithms
- Rewrite or rearrange first
- Can't solve exactly
Student Solutions
Thank you to everyone who has submitted solutions to this problem.
Here are Sergio from Kings College of Alicante's solutions to these equations.
Kathryn from Sandbach High School has also solved these equations, and has reflected on what methods she used:
I used a variety of methods to solve the equations including logs to a base, taking natural logs. Some required rearranging, others substituting to find a quadratic. The graph of $y=(5^{2x})−(5^x)−6$ was rather unusual. The 2 equations similar to $5^x +4^x =8$ were the equations I couldn't solve exactly.
Can anyone think of a nice way of grouping these equations (perhaps in a venn diagram) according to methods used to solve them?