Unit 3 Section 1 and Section 2

Unit 3 Section 1 and Section 2 - Unit 3 Exponential and...

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Unit 3 Exponential and Logarithmic Functions In the previous unit, we discussed algebraic functions, their domains, ranges and graphs. For the next two units, we will be looking at functions which are not algebraic, called transcendental functions. For this unit, we will consider a pair of one-to-one transcendental functions, which are inverses of each other. These are the exponential and logarithmic functions. We will also consider their domains, ranges and graphs. In Section 3, we will go back to solving equations. This time, the equations will involve exponential and logarithmic expressions. In the last section, we will also have a glimpse of another set of transcendental functions, called the hyperbolic functions. In this Unit, we aim to do the following: 1. Identify properties and sketch graphs of exponential functions, 2. Identify properties and sketch graphs of logarithmic functions, 3. Solve equations involving exponential and logarithmic functions, 4. Perform operations on hyperbolic and inverse hyperbolic functions, and 5. Solve verbal problems involving exponential and logarithmic functions. 3.1 Exponential Functions: Properties and Graphs Consider this problem: An amount of PhP 5,000 is deposited at 6% interest compounded annually. Find the total amount after 3 years. Let us observe at what happens to the amount invested in the table below. Year Amount invested Interest Earned Total Amount 1 5000 300 5300 2 5300 318 5618 3 5618 337.08 5955.08 Note that every year the amount invested increases because of the interest earned in that year. Thus, every year the interest earned also increases. Now, let us give an expression that will give us the total amount in t years, where t is any natural number. If P is the amount invested and i is the interest rate (in decimal form), then we have . In this expression, the variable for time is the exponent, unlike in the previous expressions that we have seen where we have constants as exponents. An expression like this is called an exponential expression.

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Unit 3. Exponential and Logarithmic Functions page 2 Let us first look at exponential functions and their properties. Definition If and , then the exponential function with base b is defined by . Illustration 1 Let us consider the function defined by . Since the variable is the exponent, the function values will just give us various powers of 2. Observe the following function values when x is an integer shown on the table on the left. 0
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This note was uploaded on 05/13/2011 for the course MATH 17 taught by Professor Dikopaalam during the Spring '11 term at University of the Philippines Los Baños.

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Unit 3 Section 1 and Section 2 - Unit 3 Exponential and...

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