Presentation on theme: "Section 6.4 Solving Logarithmic and Exponential Equations"— Presentation transcript:
1 Section 6.4 Solving Logarithmic and Exponential Equations
2 Suppose you have $100 in an account paying 5% compounded annually. Create an equation for the balance B after t yearsWhen will the account be worth $200?
3 In the previous example we needed to solve for the input Since exponential functions are 1-1, they have an inverseThe inverse of an exponential function is called the logarithmic function or logIn other wordsIf b = 10 we have the common log
4 ExampleRewrite the following expressions using logs
5 Logarithms are just exponents Evaluate the following without a calculator by rewriting as an exponential equation:
6 Logarithms are inverses of exponential functions so To see why this works rewrite the logarithm as an exponential equationAlsoTo see why this works rewrite the exponential equation as a logarithmic equationEvaluate
7 Properties of the Logarithmic Function Now the log we have in our calculator is the common log so b = 10. There is also the natural log, ln on our calculator where b = e. It has all the same properties.
9 Evaluate without a calculator Simplify without a calculator
10 Change of Base FormulaIt turns out we can write a log of a base as a ratio of logs of the same baseThis is useful if our solution contains a log that does not have a base of 10 or eThe Change of Base Formula isFor us we typically use 10 or e for a since that is what we have in our calculator
11 Reminder: The half-life of a substance is the amount of time it takes for a decreasing exponential function to decay to half of its initial valueThe half-life of iodine-123 is about 13 hours. You begin with 100 grams of iodine-123.Write an equation that gives the amount of iodine remaining after t hoursHint: You need to find your rate using the half-life informationDetermine the number of hours for your sample to decay to 10 grams
12 Reminder: Doubling time is the amount of time it takes for an increasing exponential function to grow to twice its previous levelWhat is the doubling time of an account that pays 4.5% compounded annually? Quarterly?Recall that the population of Phoenix went up by 45.3% between Assuming that growth remained steady, what is the doubling time of the Phoenix population?
13 Any exponential function can be written as Q = abt or Q = aekt Then b = ekt so k = lnbConvert the function Q = 5(1.2)t into the form Q = aektWhat is the annual growth rate?What is the continuous growth rate?Convert the function Q = 10(0.81)t into the form Q = aektWhat is the annual decay rate?What is the continuous decay rate?
14 Let’s try a few from the chapter 6.4 – 1, 3, 5, 7, 9, 19, 27, 29, 37, 49