Textbook Question
Use properties of logarithms to rewrite each function, then graph. ƒ(x) = log2 [4 (x-3) ]
Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Chapter 5, Problem 102
To solve each problem, refer to the formulas for compound interest. A = P (1 + r/n)tn and A = Pert Find t, to the nearest hundredth of a year, if \$1786 becomes \$2063 at 2.6%, with interest compounded monthly.
Verified step by step guidance1
Identify the given values: the principal amount \(P = 1786\), the amount after time \(A = 2063\), the annual interest rate \(r = 2.6\% = 0.026\), and the compounding frequency \(n = 12\) (monthly).
Write down the compound interest formula for interest compounded periodically: \(A = P \left(1 + \frac{r}{n}\right)^{nt}\).
Substitute the known values into the formula: \(2063 = 1786 \left(1 + \frac{0.026}{12}\right)^{12t}\).
Divide both sides by 1786 to isolate the exponential term: \(\frac{2063}{1786} = \left(1 + \frac{0.026}{12}\right)^{12t}\).
Take the natural logarithm of both sides to solve for \(t\): \(\ln\left(\frac{2063}{1786}\right) = 12t \cdot \ln\left(1 + \frac{0.026}{12}\right)\). Then, solve for \(t\) by dividing both sides by \(12 \cdot \ln\left(1 + \frac{0.026}{12}\right)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Compound Interest Formula
The compound interest formula calculates the amount A accumulated over time with principal P, interest rate r, number of compounding periods n per year, and time t in years. It is given by A = P(1 + r/n)^(nt), which accounts for interest being added periodically, causing exponential growth.
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Solving for Time in Exponential Equations
To find the time t in compound interest problems, you often need to solve an exponential equation. This involves isolating the exponential term and using logarithms to solve for t, since t appears as an exponent in the formula.
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Interest Rate and Compounding Frequency
The interest rate r must be expressed as a decimal, and the compounding frequency n affects how often interest is added. Monthly compounding means n = 12, which influences the growth rate and the calculation of time needed for an investment to reach a certain amount.
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Related Practice
Textbook Question
Work each problem. Which of the following is equivalent to ln(4x) - ln(2x) for x > 0? A. 2 ln x B. ln 2x C. (ln 4x)/(ln 2x) D. ln 2
Textbook Question
To solve each problem, refer to the formulas for compound interest. A = P (1 + r/n)tn and A = Pert At what interest rate, to the nearest hundredth of a percent, will \$16,000 grow to \$20,000 if invested for 7.25 yr and interest is compounded quarterly?
Textbook Question
Use properties of logarithms to rewrite each function, then graph. ƒ(x) = log3 [9 (x+2) ]
Textbook Question
Use properties of logarithms to rewrite each function, and describe how the graph of the given function compares to the graph of g(x) = ln x. ƒ(x) = ln(e2x)
Textbook Question
Use properties of logarithms to rewrite each function, then graph. ƒ(x) = log3 (x+1)/9