Proving the laws of logarithms
Here you have an opportunity to explore the proofs of the laws of logarithms.
Problem
This resource is from Underground Mathematics.
Warm up
Take a look at these results
$$\log_3 2 + \log_3 5 = \log_3 10$$
$$\log_2 15 - \log_2 3 = \log_2 5$$
$$2\log_5 7 = \log_5 49$$
$$\frac{1}{3} \log_5 64 = \log_5 4$$
$$(\log_5 7) \times (\log_7 11)=\log_5 11$$
How can you change the input values so that the equations still hold?
What happens if you change the base of the logarithms?
Can you state generalised versions of these results? What conditions must the base and the inputs satisfy?
Main problem
These cards can be sorted to give a proof of the statement
Can you arrange the cards to give a convincing proof?
You may need to include some additional algebraic steps or explanations if you think they would help to make the argument clearer or more convincing.
You might want to print out the cards and rearrange them. Some blank cards have been included in the cards for printing in case you would like to use them to fill in some details.
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
Teachers' Resources
Why use this resource?
The resource starts with a Warm-up where students are encouraged to generalise some statements about logarithms and observe important connections. They will go on to prove these results in the main parts of the resource. Students are supported to prove the first result or law using a skeleton Proof sort and then adapt this approach to prove the remaining results.
The skeleton proof in the Proof sort section includes some commentary on the steps in the proof as well as formal steps in algebra. There are also blank cards which students could fill in to include some extra steps in the algebra or explain more of the thinking behind the steps.
By working through the proof for themselves students will gain better understanding of where the log laws come from and why they are true.
Preparation
The cards should be prepared. You may want one set per student depending on the approach taken.
Possible approach
Use of the Warm-up will depend on students’ prior experience with logarithms. It may help students to recall log laws that they have already encountered. If students haven’t already seen log laws, the relationships between the numbers in these particular examples may give them a way in to seeing a general form. Students can vary the numbers in the examples and use a calculator to test whether the equations still hold, but they should also be encouraged to think about why these results make sense from the definition of a logarithm.
To prove the first generalised result, students may like to try the card sort on their own first and then compare their proofs to see which extra steps others have included. They should be encouraged to question each other to deepen their understanding and help them think about what extra steps could be helpful - should these be algebraic steps or extra explanation?
Key questions
Warm-up
Can you describe relationships between the numbers in the equations? Test the result for other numbers that have this relationship or do not have this relationship.
What if we had general inputs, e.g. $a$ and $b$ instead of $2$ and $5$?
Proof sort
What is the result you are trying to prove?
Do all your steps follow on logically from the previous ones?
What extra help might someone reading the proof need to move from one statement to the next?
Do the extra steps you have included read as complete (mathematical) sentences?
Adapting the ideas
How did the steps in the proof sort help you to prove that result?
What’s the same and what’s different about the results you’re trying to prove?
Possible support
If using the Warm-up, students may need to be encouraged to think about what it means to generalise a result. The questions at the bottom of the page may support this. For example, what would happen if they changed the $5$ in the first equation to a $7$? Do they think the equation would still hold? If not, how could they adapt the other numbers to make the equation hold. Use of a calculator or spreadsheet should support these investigations.
Some students might need help in using either words or symbols accurately and helpfully. Thinking about what the word "therefore" is there for will help students to understand its position in a proof.
Possible extension
The students can be asked to adapt the ideas in this proof to help them prove the other laws of logarithms. Students can check each others’ proofs and question each other to help decide what steps are needed.