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**Section 6.4 Solving Logarithmic and Exponential Equations**

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**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?

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**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

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Example Rewrite the following expressions using logs

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**Logarithms are just exponents**

Evaluate the following without a calculator by rewriting as an exponential equation:

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**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

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**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.

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The Natural Logarithm

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**Evaluate without a calculator**

Simplify without a calculator

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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

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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

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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 Assuming that growth remained steady, what is the doubling time of the Phoenix population?

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**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?

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**Let’s try a few from the chapter**

6.4 – 1, 3, 5, 7, 9, 19, 27, 29, 37, 49

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