Solving Exponential and Logarithm Equations

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Presentation transcript:

Solving Exponential and Logarithm Equations PreCalculus 3-4

Exponential & Log Equations

Exponential & Log Equations

Exponential & Log Equations Find the solution to the equation, correct to three decimal places x=-1.535 Exponential & Log Equations

Exponential & Log Equations

Exponential & Log Equations Solve the following equation, correct to three decimal places x=0.549 Exponential & Log Equations

Exponential & Log Equations

Exponential & Log Equations

Exponential & Log Equations Solve the equation. x=1.951 Exponential & Log Equations

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Exponential & Log Equations Solve the equation. x=0.693 Exponential & Log Equations

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Exponential & Log Equations Solve the equation. x=0 x=5 Exponential & Log Equations

Exponential & Log Equations

Exponential & Log Equations

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Exponential & Log Equations Solve the equations x=20.086 x=14 Exponential & Log Equations

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Exponential & Log Equations Solve the equation x=16666.667 Exponential & Log Equations

Exponential & Log Equations

Exponential & Log Equations

Exponential & Log Equations Solve the equation x=2 Exponential & Log Equations

Exponential & Log Equations Solve the equation x=1001 Exponential & Log Equations

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Exponential & Log Equations Solve the equation graphically x=-3.101 x=5.997 Exponential & Log Equations

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Exponential & Log Equations Exploration How long is it going to take to double my money? Exponential & Log Equations

How long is it going to take to double my money? Consider an investment of $100 invested at 5%, compounded continuously. How long would it take for the investor to have $200?

How long is it going to take to double my money? What would the doubling time be if the initial investment were $1,000? $10,000? What effect does changing the principal have on the doubling time, and why? The doubling time is not affected by changes in principal. Algebraically, the Po drops out of the exponential growth equation. Intuitively, the doubling time is a property of the ratio of two numbers, not a property of the numbers themselves.

How long is it going to take to double my money? One of the first things that is taught in an economics class is the Rule of 72. It can be summarized: "The number of years it takes an investment to double is equal to 72 divided by the annual percentage interest rate."

How long is it going to take to double my money? What would the Rule of 72 say the doubling time of a 5% investment is? Is it a good estimate? 14.4 years 3.87% error

How long is it going to take to double my money? Repeat for an investment of $100 at an interest rate of 3%, 8%, 12% and 18%. What can you say about the accuracy of the Rule of 72? Interest Rate Actual Doubling Time Estimated Doubling Time Error 3% 23.1 years 24 years 3.90% 8% 8.66 years 9 years 3.92% 12% 5.78 years 6 years 3.81% 18% 3.85 years 4 years

How long is it going to take to double my money? Derive a precise formula for the time T to double an initial investment.

How long is it going to take to double my money? There is an integer that gives a more accurate answer for continuous or nearly continuous compounding than the Rule of 72. What is this number? Check your answer by using it to estimate the doubling time of a 5% investment. 69 13.8 years 0.4% error

How long is it going to take to double my money? It turns out that there is a reason that we use the number 72 in the Rule. It has to do with one of the assumptions we made. Why do economists use the Rule of 72? If the compounding is assumed to be annual, 72 works best for values of r between 3% and 15%. For example, 10% compounded annually doubles every 7.2735 years. The rule of 72 gives 7.2, a good approximation, and the rule of 69 gives 6.9, a bad one.

Exponential & Log Equations Homework pg. 213 - 215 18, 24, 28, 32, 34, 38, 48, 50, 52, 58, 76, 80, 82, 84, 103, 106, 109, 110, 112, 115, 121 Exponential & Log Equations