4.2: Properties of Logarithms Part II

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Objectives:

• Understand and practice using common log and natural log
• Discover and utilize properties of logarithms
• Use properties of logarithms to solve exponential equations.
• Use Properties of exponentials to solve logarithmic equations.

Important Items

This lesson is a continuation of 4.2: Properties of Logarithms Part I

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Have students work on problem 1 as a sort of warm-up. You may need to remind them of some of the properties as they work.

Do some examples similar to Problems 2 In your examples, it is imperative that you use language that enforces that logs are not something like a number you multiply by, it's a function you compose with both sides. Saying something as literal as "And now we compose both sides with log by plugging both sides into log" would be productive. Otherwise, students may do things like distribute the word 'log' or divide the word alone to the other side.

-Example:

-Example:

Have students do Problem 2.

Have students work on problem 3. Note: after asking them to work on it for 3 minutes you should bring everyone together and briefly lead a discussion to get 6e^{0.012t}=8 written on the board.

Have students work on problem 4. Be prepared to remind a few tables of the proper equations to use.

Comments

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• Note that for problem 2e, $x>0$. Be aware that if students naively apply the properties they might get to $x^4 = 10^4$ and conclude $x = \pm 100$.
• Students tend to struggle to understand logarithms as a function. Be very careful with the language you use while solving for an input using a log. Saying something like "apply log to both sides" might come off as "multiply by log on both sides" and this will cause students to lose all understanding of log as a function. To enforce the function aspect, I suggest saying "compose both sides with the logarithm function." In examples, it may be better to avoid what the workbook does, and do something like "Evaluate log_3(x) at x=9", to emphasize that log is really a function.
• Instead of doing problems similar to problem 1 in 8.2, it may better to do an example: "What is 10^log(100)?" Walking them through this may more naturally lead them to see that log(100) is the power of 10 that gives 100.