Why the Power Rule/Change of Base Rule

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Presentation transcript:

Why the Power Rule/Change of Base Rule Let Then So raising to the rth power yields Therefore “the log of x to the rth power is r times the log of x” Again Or exponent

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

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.

Interest compounded k times per year If 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

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

e and compounding continuously Recall: 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

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 period This simplifies to Find the APV for $1000 at 8% interest compounded Quarterly! Monthly! Weekly! Daily! Continuously !

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?