To log or not to log?
Which of these logarithmic challenges can you solve?
Problem
This resource is from Underground Mathematics.
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?
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
Alternatively, since some equations may come under more than one heading, you could draw this as a Venn diagram.
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?
Teachers' Resources
Why use this resource?
This is a sorting exercise requiring students to think about which tools they would use to solve a range of equations involving exponentials of one sort or another. Sometimes the use of logarithms is essential, sometimes it is one method amongst others and sometimes it will not help at all.
Preparation
The set of equations to sort is available as a separate printable that could be printed on card or laminated for reuse if appropriate.
Each group of students should be given a set of the cards. Before starting, they should be encouraged to think about and discuss the headings under which they could sort the equations. To start off this discussion they might be encouraged to look at one or two carefully chosen equations that demonstrate the differences.
Possible approach
Students are probably best tackling this task in groups of two or three so that they can check one another’s work as they go.
After working at it briefly, they could be asked to share with the class what categories they are using and given opportunity to refine what they are doing.
There is a section allowing the cards to be sorted on screen which could be used at this initial sharing, during the activity or for final plenary. Headings or Venn diagrams could be drawn if displayed on a board.
Note that students are not required to actually solve each equation, though many of them will often be solved in the process of deciding what methods are required.
Key questions
- What is the same and what is different about that pair of equations?
- Can you explain what it is about that equation that makes it go under that heading?
- Can you think of a different way of solving that equation which does/doesn’t involve using logarithms?
Possible support
Students can be encouraged to start with the simpler and more familiar equations. Ensure they understand the methods they use and how to describe those methods.
Possible extension
Students could be asked to write further examples of equations for each heading. They could also think about alternative categorisations.
If a Venn diagram has been used, are there any empty regions? How could they be filled in?
Students could explore possible methods for solving the equations for which analytical methods aren’t available.