In this activity, students compare logarithmic expressions as they play a game of “war” in groups of 2. Students receive cards containing logarithms in bases 2, 5, and 10. The numbers in the expressions are written in all forms, including decimals, fractions, whole numbers, and exponential expressions. To compare the logarithms on the cards, they may:

• Determine if an expression is negative, zero, or positive, without finding the exact or estimated value.
• Estimate the value of an expression such as by using logarithms with integer values.
• Convert decimals to fractions to help find the logarithms.

As students preview the deck by putting the cards in order from least to greatest, listen for how they decide on the relative values of the cards. Students should not use calculators at this point.

If students do not yet correctly make estimates to compare expressions, consider asking:

• “How did you decide which card value is greater?”

• “How does each expression compare to 0? To 1? How do you know?”

Much of the discussions will happen in groups, as students put the cards in order before playing the game and while playing the game. Invite groups to share any insightful or useful strategies for comparing any two cards that came up before or during the game.

This activity has two aims: to familiarize students with the log function on a calculator, and to reinforce the fact that most values obtained on a calculator, like those in a log table, are approximations.

The questions involve finding only base-10 logarithms. This is partly to focus the exploration, and partly because many calculators can evaluate only logarithms in base 10 and base , or require the user to perform steps to change the base before calculating logarithms in other bases. Some calculators allow easily calculating logs with any base. If the devices used in your class allow for this, show students how it works.

The purpose of the last question is to prompt students to think about reasonable values of the logarithms and to not simply enter numbers into the calculator. Monitor for students with clear estimation strategies. Beyond reasoning that a log value is between two whole numbers, students don't yet have a good means of making an accurate estimate, which is fine. For example, they might be able to say that should be quite close to 2, but it is not expected that they could say how close.

Make sure students understand how the log function on their calculator works and can find log values correctly.

Invite previously selected students to share how they estimated log values before using a calculator. Emphasize that having a sense of what value would be reasonable before entering a logarithm in a calculator can help us notice any errors.