Lesson 4 Log-Arithm-etic Practice Understanding
Graph each of the following functions:
Use logarithm properties to find equivalent algebraic expressions.
Use logarithm properties to find values for logarithmic expressions using other known values.
How can the properties of logarithms help us in working with logarithmic expressions?
Open Up the Math: Launch, Explore, Discuss
Abe and Mary are feeling good about their log rules and bragging about their mathematical prowess to all their friends when this exchange occurs:
Stephen: I guess you think you’re pretty smart because you figured out some log rules, but I want to know what they’re good for.
Abe: Well, we’ve seen a lot of times when equivalent expressions are handy. Sometimes when you write an expression with a variable in it in a different way, it means something different.
What are some examples from your previous experience where equivalent expressions were useful?
Mary: I was thinking about the Log Logic task where we were trying to estimate and order certain log values. I was wondering if we could use our log rules to take values we know and use them to find values that we don’t know.
For instance: Let’s say you want to calculate
Stephen: That’s great. Everyone knows that
Abe: Oh, I saw this somewhere. Uh,
Based on what you know about logarithms, explain why
Stephen: Oh, oh! I’ve got one. I can figure out
Can Stephen and Mary both be correct? Explain who’s right, who’s wrong (if anyone) and why.
Now you can try applying the logarithm rules yourself. Use the values that are given and the ones that you know by definition, like
The three rules, written for any base
Logarithm of a Product Rule:
Logarithm of a Quotient Rule:
Logarithm of a Power Rule:
Given the work you have just done, what other values would you need to figure out the value of the base
Sometimes thinking about equivalent expressions with logarithms can get tricky. Consider each of the following expressions and decide if they are always true for the numbers in the domain of the logarithmic function, sometimes true, or never true. Explain your answers. If you answer “sometimes true,” describe the conditions that must be in place to make the statement true.
Ready for More?
You have already figured out values like
Create some more logarithmic expressions that can be calculated. Challenge another student with your logarithmic expression and then try to figure out the logarithmic expression that they created.
Strategies for working with logarithmic expressions and using properties of logarithms:
In this lesson, we learned to find the value of a logarithmic expression combining known values and using the logarithm properties. We determined if certain logarithmic equations were true and reinforced that the logarithm product and quotient rules are only for multiplication and division inside the argument of the logarithm.
Factor out the greatest common factor in the expression. Then rewrite the numbers inside the parentheses.
Use your calculator and the definition of