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Unit 3 Section 3

# Unit 3 Section 3 - 3.3 Equations Involving Exponential and...

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3.3 Equations Involving Exponential and Logarithmic Expressions In this section, we will consider more examples on modeling real life problems, but this time we will use exponential functions. We will also discuss how to solve these problems and other equations that involve exponential and logarithmic functions. Natural Exponential Function Recall from Section 1 our sample problem: An amount of PhP 5,000 is deposited at 6% interest compounded annually. Find the total amount after 3 years. We represented the total amount in t years as , where P is the amount invested and i is the interest rate (in decimal form). Generally, we can use the following function to model the total amount (or accumulated value) at any time t , denoted by . where P is the amount invested (called the principal), i is the interest rate (in decimal form) and m is the number of times in a year the amount is compounded. Note that in the given problem, the amount in compounded annually, then it is compounded once a year or . In that case, . Illustration 1 Suppose a principal worth PhP 100000 is invested for 10 years at interest rate 6%. At a simple interest rate, the accumulated value is given by , computed to be equal to PhP 160000. Consider the following table that shows the accumulated values when the interest rate is compounded in various ways. Interest Accumulated Value simple PhP 160000 compounded annually PhP 179084.77 compounded semi- annually PhP 180611.12 compounded monthly PhP 181939.67 compounded weekly PhP 182148.87 compounded daily PhP 182202.90

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Unit 3. Exponential and Logarithmic Functions page 2 Is it possible for the principal amount to be compounded continuously? Let us see what happens if we increase m without bound. Observe the expression . This can also be written as . If we let , we get . Since the value for i is fixed, as we increase m without bound, the value of a also increases without bound. Let us focus on the expression , and see what happens as we increase the value of a without bound. Observe the following table. a B 1 2 2 2.25 4 2.44140625 10 2.5937424601 100 2.7048138294215260932671947108075 1000 2.7169239322358924573830881219476 10000 2.7181459268252248640376646749131 100000 2.718268237174489668035064824426 As we continue to increase the value of a , we will see that the value of B approaches a certain irrational number, which is denoted by e. Therefore, if the amount invested is compounded continuously, the accumulated
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Unit 3 Section 3 - 3.3 Equations Involving Exponential and...

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