Textbook Question
Determine whether each function graphed or defined is one-to-one. y = ∛(x+1) - 3
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Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 5, Problem 33
Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. 5(1.015)x-1980 = 8
Verified step by step guidance1
Start with the given equation: \(5(1.015)^{x-1980} = 8\).
Isolate the exponential expression by dividing both sides of the equation by 5: \((1.015)^{x-1980} = \frac{8}{5}\).
To solve for the exponent \(x - 1980\), take the natural logarithm (ln) of both sides: \(\ln\left((1.015)^{x-1980}\right) = \ln\left(\frac{8}{5}\right)\).
Use the logarithm power rule to bring down the exponent: \((x - 1980) \cdot \ln(1.015) = \ln\left(\frac{8}{5}\right)\).
Finally, solve for \(x\) by dividing both sides by \(\ln(1.015)\) and then adding 1980: \(x = \frac{\ln\left(\frac{8}{5}\right)}{\ln(1.015)} + 1980\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Equations
An exponential equation involves variables in the exponent, such as a^(x) = b. Solving these requires isolating the exponential expression and often using logarithms to solve for the variable in the exponent.
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Solving Exponential Equations Using Logs
Logarithms and Their Properties
Logarithms are the inverse operations of exponentials, allowing us to solve equations where the variable is an exponent. Key properties include log(a^b) = b*log(a), which helps isolate the variable when taking logarithms of both sides.
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5:36Change of Base Property
Rounding Decimal Answers
When solutions are irrational, they are often expressed as decimals rounded to a specified place value. Here, answers must be rounded to the nearest thousandth, meaning three digits after the decimal point.
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4:47The Number e
Related Practice
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Textbook Question
Solve each equation. log2 x = 3
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Textbook Question
Determine whether each function graphed or defined is one-to-one. y = 5|x+2|
Textbook Question
Use a calculator to find an approximation to four decimal places for each logarithm. log2/3 5/8