4.5 Apply Properties of Logarithms
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4.5 Apply Properties of Logarithmsp. 259
What are the three properties of logs?How do you expand a log? Why?How do you condense a log?
Properties of Logarithms
Use log53.683 and log571.209log521 =log5(37)=log53 + log57.683 + 1.209 =1.892
Use log53.683 and log571.209Approximate:log549 =log572 =2 log57 2(1.209)=2.418
Write 40 as 8 5.Product propertySimplify.
Expanding LogarithmsYou can use the properties to expand logarithms.
log27x3 - log2y = log27 + log2x3 log2y =log27 + 3log2x log2y
Your turn!Expand:log 5mn =log 5 + log m + log n
Expand:log58x3 =log58 + 3log5x
Condensing Logarithmslog 6 + 2 log2 log 3 =log 6 + log 22 log 3 =log (622) log 3 =
SOLUTIONEvaluate using common logarithms and natural logarithms.Using common logarithms:Using natural logarithms:
What are the three properties of logs?Productexpanded add each, Quotientexpand subtract, Powerexpanded goes in front of log.How do you expand a log? Why?Use logb before each addition or subtraction change. Power property will bring down exponents so you can solve for variables.How do you condense a log?Change any addition to multiplication, subtraction to division and multiplication to power. Use one logb
For a sound with intensity I (in watts per square meter), the loudness L(I) of the sound (in decibels) is given by the function
Product propertySimplify.SOLUTIONLet I be the original intensity, so that 2I is the doubled intensity.Increase in loudnessWrite an expression.Substitute.Distributive propertyUse a calculator.
4.5 Assignmentpage 262, 7-41 odd
Properties of Logarithms Day 2What is the change of base formula?What is its purpose?
Your turn!Condense:log57 + 3log5t =log57t3
Condense:3log2x (log24 + log2y)=
Change of base formula:a, b, and c are positive numbers with b1 and c1. Then:
logca = (base 10)
logca = (base e)
Examples:Use the change of base to evaluate:log37 =(base 10)log 7 log 31.771(base e)ln 7 ln 31.771
Use the change-of-base formula to evaluate the logarithm.SOLUTIONSOLUTION
What is the change of base formula?
What is its purpose?Lets you change on base other than 10 or e to common or natural log.
4.5 Assignment Day 2Page 262, 16- 42 even, 45-59 odd