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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Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 5, Problem 85
In Exercises 83–88, let logb 2 = A and logb 3 = C and Write each expression in terms of A and C.
logb 8
Verified step by step guidance1
Recognize that the expression log_b 8 can be rewritten using the fact that 8 is a power of 2. Since 8 = 2^3, rewrite the logarithm as log_b (2^3).
Apply the logarithmic power rule, which states that log_b (x^n) = n * log_b x. Using this, rewrite log_b (2^3) as 3 * log_b 2.
Recall the given information that log_b 2 = A. Substitute this into the expression to get 3 * A.
Thus, the expression log_b 8 is written in terms of A as 3A.
No further simplification is needed since the problem asks to express the logarithm in terms of A and C.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithm Properties
Logarithms have specific properties such as the product, quotient, and power rules. The power rule states that log_b(x^n) = n * log_b(x), which allows us to simplify expressions like log_b(8) by expressing 8 as a power of a base number.
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Change of Base Property
Change of Base and Given Logarithms
When given specific logarithmic values like log_b(2) = A and log_b(3) = C, we can rewrite other logarithms in terms of these known values by expressing the argument as products or powers of 2 and 3. This helps in expressing complex logs in simpler terms.
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Prime Factorization
Prime factorization involves expressing a number as a product of prime numbers. For example, 8 can be written as 2^3. This factorization is essential to rewrite log_b(8) in terms of log_b(2), which is given as A, enabling simplification using logarithm properties.
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Related Practice
Textbook Question
Evaluate or simplify each expression without using a calculator. In 1
Textbook Question
Evaluate or simplify each expression without using a calculator. 10log 33
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+log(x+3)=log 10
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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. 2 log x−log 7=log 112
Textbook Question
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?
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