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Section 6.4 Solving Logarithmic and Exponential Equations

Section 6.4 Solving Logarithmic and Exponential Equations. Suppose you have $100 in an account paying 5% compounded annually. Create an equation for the balance B after t years When will the account be worth $200?. In the previous example we needed to solve for the input

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Section 6.4 Solving Logarithmic and Exponential Equations

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  1. Section 6.4Solving 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 years • When 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 inverse • The inverse of an exponential function is called the logarithmic function or log • In other words • If b = 10 we have the common log

  4. Example • Rewrite 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 equation • Also • To see why this works rewrite the exponential equation as a logarithmic equation • Evaluate

  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.

  8. The Natural Logarithm

  9. Evaluate without a calculator Simplify without a calculator

  10. Change of Base Formula • It turns out we can write a log of a base as a ratio of logs of the same base • This is useful if our solution contains a log that does not have a base of 10 or e • The Change of Base Formula is • For 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 value • The 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 hours • Hint: You need to find your rate using the half-life information • Determine 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 level • What 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 1990-2000. 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 = lnb • Convert the function Q = 5(1.2)t into the form Q = aekt • What is the annual growth rate? • What is the continuous growth rate? • Convert the function Q = 10(0.81)t into the form Q = aekt • What 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

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