Ch. 4 - Exponential and Logarithmic Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

Problem 85

Chapter 5, Problem 85

Use the formula for continuous compounding to solve Exercises 84–85. What annual rate, to the nearest percent, is required for an investment subject to continuous compounding to triple in 5 years?

Verified step by step guidance

Identify the formula for continuous compounding: $ A = P e^{rt} $, where $ A $ is the final amount, $ P $ is the principal (initial investment), $ r $ is the annual interest rate (in decimal form), $ t $ is the time in years, and $ e $ is the base of the natural logarithm.

Substitute the given values into the formula. Since the investment triples, $ A = 3P $, $ t = 5 $, and $ P $ is the initial principal. The equation becomes $ 3P = P e^{5r} $.

Simplify the equation by dividing both sides by $ P $ (assuming $ P \neq 0 $): $ 3 = e^{5r} $.

Take the natural logarithm ($ \ln $) of both sides to isolate $ r $: $ \ln(3) = 5r $.

Solve for $ r $ by dividing both sides of the equation by 5: $ r = \frac{\ln(3)}{5} $. This gives the annual rate in decimal form. To express it as a percentage, multiply the result by 100.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Continuous Compounding

Continuous compounding refers to the process of earning interest on an investment at every possible moment, rather than at discrete intervals. The formula used for continuous compounding is A = Pe^(rt), where A is the amount of money accumulated after time t, P is the principal amount, r is the annual interest rate, and e is the base of the natural logarithm. This method allows for the maximum growth of an investment over time.

Exponential Growth

Exponential growth occurs when the growth rate of a value is proportional to its current value, leading to rapid increases over time. In the context of continuous compounding, the investment grows exponentially as interest is calculated continuously. This concept is crucial for understanding how investments can increase significantly over a relatively short period, especially when compounded continuously.

Natural Logarithm

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is used in continuous compounding to solve for the time or rate when dealing with exponential equations. In this scenario, the natural logarithm helps to isolate the variable r (the annual interest rate) when determining how long it takes for an investment to reach a certain value under continuous compounding.

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Related Practice

Textbook Question

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log(x+4)−log 2=log(5x+1)

Textbook Question

In Exercises 83–88, let log_b 2 = A and log_b 3 = C and Write each expression in terms of A and C.

log_b (3/2)

Textbook Question

Use the formula for continuous compounding to solve Exercises 84–85. How long, to the nearest tenth of a year, will it take \$50,000 to triple in value at an annual rate of 7.5% compounded continuously?

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Textbook Question

In Exercises 83–88, let log_b 2 = A and log_b 3 = C and Write each expression in terms of A and C.

log_b 8

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Textbook Question

Evaluate or simplify each expression without using a calculator. 10^{log 33}

Textbook Question