Title The Properties of Logarithms Overview of Lesson
Students will simplify and evaluate expressions using the properties of logarithms. Students will solve logarithmic equations using the properties of logarithms. Prerequisites
Graphing exponential functions Solving exponential equations and inequalities
Logarithms and Logarithmic Functions Solving logarithmic equations and inequalities
Learning Targets
Review 7.1-7.4 Use the properties of logs to expand and condense equations Evaluate common logs Time Required 90 minutes Materials Required Flash drive, white boards, markers, erasers, notes page Instructional Instructional Lesson Plan a. Bacteria population grown: Original population 6000 bacteria cells. After 2 hours, there were 28,000 cells. ce lls. Write an exponential function that could be used to model the number of bacteria after x hours. b. Review (white boards) i. Write in exponential form: Log381 = 4 (34 = 81) ii. Write in logarithmic form: 9 3 = 729 (log9729 = 3) iii. Evaluate log5125 2 c. Caroline Interview i. My roommates use logs ii. 1:33 Write equation iii. In chemistry, there are formulas using log to find the pH (or level of acidity) of different substances. This is important for consumers with sensitive stomachs that are looking to cut back on acidic ac idic foods. A low pH indicates an acidic solution and a high pH indicates a basic solution. The pH scale is based o n powers of ten, so we can find out that coffee which has a pH of 5 is 100 times as acidic as water which has a pH of ten because 107-5 = 102 = 100. d. Basic Log rules/properties i. Simplifying expressions with exponents ii. Product Property 1. We’ve simplified expressions with exponents when we multiplied x 5 by x7. How can I condense this expression? Add the exponents to get x12 2. On your white board simplify the expression x m ∙ xn xm+n This is the product property. We can use it with logarithms as well.
worksheet
log b(mn) = log b(m) + log b(n) This allows us to expand logarithmic expressions with the same base. Multiplication inside the log can be turned into addition outside the log, and vice versa.
In example 3, how can we use the given value log43 ≈ .7925 to find log4192?
iii. Quotient Property 1. Just as with the product property, we have a way to simplify logarithmic expressions where there are terms that are divided within the log. When we did this with exponents, how did we condense x 17 / x11 ? x6
Worksheet
log b(m/n) = log b(m) – log b(n) Division inside the log can be turned into subtraction outside the log, and vice versa.
Ex. 4 I have division inside the log, which can be split apart as subtraction outside the log, so:
The first term on the right-hand side of the above equation can be simplified to an exact value, by applying the basic definition of what a logarithm is: log 4(16) = 2. Then the original expression expands 16
fully as: log 4(
/ x x ) = 2 – l og 4(x )
Ex. 5 turn into fractions
Ex. 7: What are we given? The formula for finding pH and the pH of the rain. What do we want to find? The amount of hydrogen in a liter of this rain. Plan: write the equation and solve for [H +]. iv. Power Property 1. An exponent on everything inside a log can be m oved out front as a
multiplier, and vice versa. n 2. log b(m ) = n · log b(m)
worksheet After Ex. 10, check the temperature of the room by having them hold up a # for how they fee l with condensing expressions. 5 means I can do this with ease, 1 means I don’t have any clue what you’re talking about.
e. Application problems
i.
Let log b(2) = 0.3869, log b(3) = 0.6131, and log b(5) = 0.8982. Using these values, evaluate log b(10).
Since 10 = 2 × 5, then log b(10) = log b(2 × 5) = log b(2) + log b(5) Since I have the values for log b(2) and log b(5), I can evaluate: log b(2) + log b(5) = 0.3869 + 0.8982 = 1.2851 Then log b(10) = 1.2851. ii. Let log b(2) = 0.3869, log b(3) = 0.6131, and log b(5) = 0.8982. Using these values, evaluate log b(9). 2
2
Since 9 = 3 , then: log 2log b(3) b(9) = log b(3 ) = 2log Since I have the value for log 2log b(3) = 2(0.6131) = 1.2262 b(3), then I can evaluate: 2log
