Start Up Day 47 1. What is the logarithmic form of 144 = 122? 2. What is the value of log927? 3. Describe the graph of y = log5x? Identify domain and range, and the “locator point” and the asymptote.
Properties of Logarithms Lesson 7-4 Properties of Logarithms
OBJECTIVE: SWBAT to use the properties of logarithms to convert between condensed and expanded form, to use the change of base formula and to evaluate logarithmic expressions. Essential Questions: How are the properties of Logarithms related to the properties of exponents? How can they be used to help evaluate logarithmic expressions? HOME-LEARNING: p. 466 #10-36 even, 47, 50-55 all, 56 & 62 Continue Test Review: p. 489 # 24, 25, 28, 29, 32, 33, 36, 38, 40, 42, 44, + MathXL online Quiz 7-1 & 7-2 (due by 2/4)
inverses “undo” each each other Since logs and exponentials of the same base are inverse functions of each other they “undo” each other. Remember that: This means that: inverses “undo” each each other = 7 = 5
Let’s Review ”The More Connections you see, the easier it will be!” Product Keep the base & ADD exponents! Quotient Keep the base & SUBTRACT the exponents! Power to a Power Keep the base & MULTIPLY the exponents!
= = = = 1. 2. 3. EXPANDED Properties of Logarithms CONDENSED (these properties are based on rules of exponents since logs = exponents)
Problem 1A: Using the log properties, write the expression as a single logarithm (condense). using the Quotient Property: this direction Simplify Evaluate, if possible—”4” to What exponent is “16”?
Problem 1B: Using the log properties, write the expression as a single logarithm (condense). using the Power Property: this direction using the Product Property: this direction
Problem 2A: Using the log properties, write the expression as a sum and/or difference of logs (expand). using the Quotient Property: using the Product Property:
Problem 2B: Using the log properties, write the expression as a sum and/or difference of logs (expand). using the Quotient Property: using the Power Property: Re-write and evaluate, if possible
More Properties of Logarithms This one says if you have an equation, you can take the log of both sides and the equality still holds. This one says if you have an equation and each side has a log of the same base, you know the "stuff" you are taking the logs of are equal.
use log property & take log of both sides (we'll use common log) There is an answer to this and it must be more than 3 but less than 4, but we can't do this one in our head. (2 to the what is 8?) Let's put it equal to x and we'll solve for x. Change to exponential form. (2 to the what is 16?) use log property & take log of both sides (we'll use common log) (2 to the what is 10?) use 3rd log property Check by putting 23.32 in your calculator (we rounded so it won't be exact) solve for x by dividing by log 2 use calculator to approximate
If we generalize the process we just did we come up with the: Example for TI-84 Change-of-Base Formula The base you change to can be any base so generally we’ll want to change to a base so we can use our calculator. That would be either base 10 or base e. “common” log base 10 LOG LN “natural” log base e
ALPHA TBLSET or F2 5 Example for TI-84 Most Calculators now have the Change-of-Base function built right in! ALPHA TBLSET or F2 5
Problem 3A: Use the Change-of-Base Formula and a calculator to approximate the logarithm. Round your answer, if necessary, to two decimal places. Did you notice that both numbers are powers of “3”? Since 81 >27, our answer of what exponent to put on 81 to get it to equal 27 will be less than 1. With out using a calculator –we could Change our Base to base “3” Or—We could use our calculator –COMMON LOG or NATURAL LOG! put in calculator
Problem 3B: Use the Change-of-Base Formula and a calculator to approximate the logarithm. Round your answer, if necessary, to two decimal places. Since 52=25 and 53=125, our answer of what exponent to put on 5 to get it to equal 36 will be between 2 and 3. Let your calculator evaluate it.
Let’s Try It! Lesson Check p. 465 Problems #1-6