Using the properties of exponents to prove the Power Property of Logarithms, unit 6 algebra homework help

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x American High School| X + o edu.americanhighschool.org/dashboards/ActiveLearning/course-website.asp?action=assignment&Submitid=27108&srloid=27108&rloid=27108&courseid=1486 = 2 American High School Dashboard * Management v O Reports Cassidy Williams Lessons Assessments Unit 6: Exponential and Logarithmic Functions End Checkpoint Apply - Unit 5: Rational Functions Project Unit 6: End Checkpoint Apply Unit 6: Exponential and Logarithmic Functions End Checkpoint Apply Interactive Tools Lesson 3-5 Algebra 2 Please complete the following questions. It is important that you show all work you did to solve the problems when you submit your work. This includes any calculations, diagrams, or graphs that helped you solve it. 1. REASONING Use the properties of exponents to prove the Power Property of Logarithms. 2. OPEN ENDED Give an example of a quantity that grows or decays at a fixed rate. Write a real-world problem involving the rate and solve by using logarithms. 3. Writing in Math Use the information about banking on page 556 to explain how the natural base e is used in banking. Include an explanation of how to calculate the value of an account whose interest is compounded continuously, and an explanation of how to use natural logarithms to find the time at which the account will have a specified value in your answer. Your Answer ... Click here to chat O Type here to search g 11:51 AM 8/14/2017 19 x American High School | CC1 S Live Discussion - Studypool CC - Algebra 2 Х + o = 22 nud Mos Tip sulat have ankey aluat Atural Arithuna. Main Ideas Essline pression in their Lesen haritis) Sche valunal impuli Ng relates intervisualtechnology.us/uploads/PDFs/ebooks/CC%20-%20Algebra%202/CC%20-%20Algebra%202 GET READY for the Lesson ne a bank compounds 556-557 / 1100 про erust accounts continewsly, that is, with no waiting time between interest payment To develop an equation to determine continuously Compounded interesitamine what happens to the value of an account increasingly larger numbers of compounding periods. Use a principal otsi, an interest rater of 100 or 1, and time of year EXCEE 100% Un calcul toe Tar og pressons apre to four decimal places 27. Calculator Keystrokes Onning Calon you pies it Netele itehner raining you musette number tebe pressing the LN Nelabulary b. In 0.0 url tes mullae டியாய் ndi!! nih anton ICH YOUR 2) In? 2.50 < Base e and Natural Logarithms in the table above, as increases the prescimen 111 + 1, "cor (1 + i spproaches the irrational mamber 71878... This number is relerred to as the natural base, You an write an equivalent baserponenti luga unic equation in VEDE Y IN Y An exponential function with base e is called a natural base exponential function. The graph of y-r' is shown at the right. Natural base exponential functions are wat extensively in since to model quantities that grow and decay coatinuously Most calculators have an e' fanction for evaluating natural base expressions EXAMPLE Write Equivalent Expressions Write an equivalent exponential or logarithmic equation. a = 5 b. In: 0.6931 - 5log 5Y In x 0.61131 - log ry 0.6931 Dit 5 = CHECK Your Progress JA.T- 38. In x l.2 Stay Tip Since the raluralhase function and the natural logarithmic functionare Invcrss, these two functions can be used to "undo' esch other. Simplifying Expressions withe You can sing restrig EXAMPLE Evaluate Natural Base Expressions O se a calculator to evaluate each expression to four decimal places. ar? KEYSTROKES: 2nd les 2 ENTER 7.38-058.99 73891 Inas For example, anda pe 13 - 4x + 3 ?: ܐܶܢ E Type here to search O x 11:57 AM 8/14/2017 719
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1. Using the properties of exponents to prove the Power Property of Logarithms:
Solution:
Power property: log 𝑏 (𝑀𝑝 ) = p × log 𝑏 (𝑀)
Proof:
1- Let: 𝑀 = 𝑏 𝑥 which gives that log 𝑏 (𝑀) = 𝑥
2- log 𝑏 (𝑀𝑝 ) = log 𝑏 ((𝑏 𝑥 )𝑝 ) (Substitution)
= log 𝑏 (𝑏 𝑥𝑝 ) (Properties of exponents)
=𝑥𝑝
( log 𝑏 (𝑏 𝑐 ) = 𝑐)
= log 𝑏 (𝑀) × 𝑝 (Substitution)
=p × log 𝑏 (𝑀 )
Another proof:
1- Let log 𝑏 (𝑀𝑝 ) = 𝑥, then 𝑏 𝑥 = 𝑀𝑝 (Equivalent exponential form)
𝑥
𝑝

𝑝
𝑝

Then 𝑏 = 𝑀 = 𝑀 (Property of exponents)
𝑥
Then log 𝑏 (𝑀) = (Equivalent log form)
𝑝

Then p × log 𝑏 (𝑀) = 𝑥 (Multiplication by p)
Then p × log 𝑏 (𝑀) = log 𝑏 (𝑀𝑝 ) (Substitution)
2. Example of a quantity that grows or decays at a fixed rate:
Money in a bank
Investment when interest rate is fixed 𝐴 = 𝑃 × (1 + 𝑟)𝑡
where: A is the final amount, P: is the principal, r: is the interest rate, t: is the
number of years.
Real-world problem:
If...


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