Let Then So raising to the r th power yields Therefore the log of x to the r th power is r times the log of x Again Or exponent Solving Exponential Equations 1 Solve done ID: 532451
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Slide1
Why the Power Rule/Change of Base Rule
LetThenSo raising to the rth power yields Therefore“the log of x to the rth power is r times the log of x”Again Or
exponentSlide2
Solving Exponential Equations
1. Solve: done!Given 50 grams of a radioactive substance with a half life of 15 days, how long until only 20 grams remain?Given an initial population of 7000 and a growth rate of 6% per year, when will the population reach 10,000?Common Logs versus Natural Logs Slide3
Solving Exponential Equations Two Ways
1. Solve two ways – first by taking the ln( ) of both sides and using log rules to simplify; second by rewriting it as and converting it to a logarithmic equation.2. If Radium has a half life of 1690 years, how old is an object if the remaining Radium is down to 30%3. If the population of Columbus OH was 632,910 in 1990 and 711,265 in 2000, find the exponential equation which models its population and determine when the population will the reach 1 million.Slide4
Interest compounded k
times per yearIf interest is compounded k times per year – divide the interest rate by k and multiply the number of years by k.Example: If you invest $1000 at 8% interest, how much do you earn after 5 years if interest is compounded Quarterly? Monthly? Weekly? Daily?How long to earn $2000 if interest is compounded Quarterly? Monthly? Weekly? Daily?Slide5
Interest Compounded Continuously
If we increase k, the number of times integers is compounded per year, we can show …If you invest $1000 at 8% interest, how much do you earn after 5 years if interest is compounded Continuously? How long to earn $2000 if interest is compounded Continuously?Slide6
e
and compounding continuouslyRecall: The limit exists and defines the value of e which is approximately 2.718281828459… If interest is compounded continuously it can be shown that To evaluate let so Slide7
APY and Present Value
A common comparison for comparing investments, Annual Percentage Yield (APY) is the percentage rate, r, that compounded annually would yield the same return as a given interest rate,r0, with a given compounding periodThis simplifies to Find the APV for $1000 at 8% interest compounded Quarterly! Monthly! Weekly! Daily! Continuously !Slide8
Present Value?
How much do you have to invest now at 8% interest compounded quarterly (monthly, weekly, daily, continuously) so that in 5 years you will have $2000?