Sec. 4.2 Logarithmic Functions

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1 Sec. 4.2 Logarithmic Functions The Logarithmic Function with Base a has domain all positive real numbers and is defined by Where and is the inverse function of So and

2 Logarithms are inverses of Exponential functions Just like square roots and cube roots are inverses of quadratic and cubic functions. Logarithms don't deserve the reputation they have for being hard or strange. They just have some rules that you can follow to simplify them or solve equations using them. Remember is the power you have to raise the base 'a' to in order to get x. If you multiply by 'a' each period, it is the number of periods it takes to reach 'x'. Logarithmic Form vs. Exponential Form

3 Properties of Log Functions Changing Bases: How can you rewrite an exponential with a different base?

4 Solve the following equations: take the Logarithm of both sides (to undo the exponential function) Graphs of Log Functions

5 There are many situations which involve exponential growth. Interest Rate Problems, Population Growth, Radioactive Decay, Newton s Law of Cooling, Absorption of Light, Atmospheric Pressure. If the function giving Population in terms of time involves an exponential function, then the inverse function giving the time when the Population is a certain value will involve a log function. To solve an equation with the variable in the exponent, will require that we take the logarithms of both sides (at some point). To solve an equation with the variable in a Log Function, will require that we exponentiate both sides (at some time). Just like, to solve an equation where the variable is squared, requires that we take square roots of both sides at some point. The time it takes for a investment of $3000 to reach a value of A dollars when it is compounded continuously at 8% annual interest is given by: When will the account be worth $6000. When will it be worth $1,000,000?

6 The time it takes for a investment of $3000 to reach a value of A dollars when it is compounded continuously at 8% annual interest is given by: What will the investment be worth in 10 years? The time it takes for a investment of $3000 to reach a value of A dollars when it is compounded continuously at 8% annual interest is given by: Find the inverse Function. That is the function that gives the amount of the investment in terms of time. The algebra is the same as in the previous problem.

7 The age of an artifact can be determined by the amount of radioactive Carbon 14 remaining in it. If D 0 is the original amount of Carbon 14 and D is the amount remaining, then the age t in years is given by Find the age on an object if 73% of the original carbon 14 remains. The age of an artifact can be determined by the amount of radioactive Carbon 14 remaining in it. If D 0 is the original amount of Carbon 14 and D is the amount remaining, then the age t in years is given by What percentage of the original carbon 14 remains after 1000 years?

8 The age of an artifact can be determined by the amount of radioactive Carbon 14 remaining in it. If D 0 is the original amount of Carbon 14 and D is the amount remaining, then the age t in years is given by Find the inverse function. That is solve the equation above for D in terms of t. Word Problems involving Exponentials Suppose you invest $5000 in an account that earns 7% interest per year. What is a function for the value of the account after t years? When will it be worth $10000?

9 Word Problems involving Exponentials Suppose you invest $5000 in an account that earns 7% interest per year compounded monthly. What is a function for the value of the account after t years? When will it be worth $10000? Suppose you invest $5000 in an account that earns 7% interest per year compounded continuously. What is a function for the value of the account after t years? When will it be worth $10000?

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