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Chapter 1: Problem 23

For Activities 19 through \(26,\) solve for the input that corresponds to each of the given output values. (Round answers to three decimal places when appropriate.) $$ r(x)=2 \ln 1.8\left(1.8^{x}\right) ; r(x)=9.4, r(x)=30 $$

Short Answer

Expert verified
For \( r(x)=9.4 \), \( x \approx 2.888 \); for \( r(x)=30 \), \( x \approx 8.794 \).

Step by step solution

01

Understand the Problem Statement

We are asked to find the input values of the function \( r(x) = 2 \ln 1.8 \times (1.8^x) \) such that the function outputs are 9.4 and 30. We will solve for \( x \) for each value of \( r(x) \).
02

Simplify the Given Function

First, simplify the function: \( r(x) = 2 \ln 1.8 + 2x \ln 1.8 \). This simplification is derived from the property of logarithms: \( \ln(a^b) = b\ln(a) \).
03

Solve for x when r(x) = 9.4

Substitute 9.4 into the simplified function: \( 9.4 = 2 \ln 1.8 + 2x \ln 1.8 \). Rearrange to solve for \( x \): \( 2x \ln 1.8 = 9.4 - 2 \ln 1.8 \). Then, solve for \( x \): \[ x = \frac{9.4 - 2 \ln 1.8}{2 \ln 1.8} \approx 2.888. \]
04

Solve for x when r(x) = 30

Substitute 30 into the simplified function: \( 30 = 2 \ln 1.8 + 2x \ln 1.8 \). Rearrange to solve for \( x \): \( 2x \ln 1.8 = 30 - 2 \ln 1.8 \). Then, solve for \( x \):\[ x = \frac{30 - 2 \ln 1.8}{2 \ln 1.8} \approx 8.794. \]
05

Verification (Optional)

Recalculate the outputs for the computed values of \( x \) and ensure they match the original outputs within rounding precision. Substituting \( x = 2.888 \) and \( x = 8.794 \) back into \( r(x) \) should give approximately 9.4 and 30, respectively.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Functions
Logarithmic functions are powerful tools in mathematics, especially when dealing with exponential relationships. They help us solve problems where the variable is an exponent, as logs allow us to simplify equations involving powers and exponents.

The key property of logarithms that we often use in calculus is: the logarithm of a power is the exponent times the logarithm of the base. This can be expressed as \( \ln(a^b) = b \ln(a) \).
This property helps to rewrite complex exponentials into simpler, more manageable algebraic expressions.
  • Natural Logarithm (ln): It is a logarithm to the base \(e\), an irrational constant approximately equal to 2.718. It simplifies calculations, especially in calculus, due to its natural properties.
  • Usage: By applying logarithms, particularly natural logs, we can break down problems involving multiplication or exponentiation into addition or multiplication problems, which are much easier to solve.
Understanding and manipulating logarithmic expressions are essential steps in simplifying equations and solving problems in calculus.
Function Simplification
Function simplification is crucial to make complex equations more manageable. In calculus, functions often contain terms that can leverage mathematical properties to reduce their complexity.
  • Using properties like \( \ln(a^b) = b \ln(a) \), we can transform parts of the function for easier handling.
  • For the function \( r(x) = 2 \ln 1.8 (1.8^x) \), simplification leads to \( r(x) = 2 \ln 1.8 + 2x \ln 1.8 \). This is a less complex and easier to interpret form.
Simplifying functions simplifies the process of equation solving. It reduces errors and streamlines calculations.

Always check for common mathematical properties that can be applied. Simplified functions not only save time but also provide clearer insight into how changes in variables affect the function overall.
Solving Equations
Solving equations is a fundamental concept in calculus, and it involves finding the values of variables that satisfy the equation. Once we have simplified an equation, solving it becomes straightforward.
  • The goal is to isolate the variable. For instance, after simplifying \( r(x) = 2 \ln 1.8 + 2x \ln 1.8 \), isolate \( x \) by rearranging terms.
  • With the equation \( 2x \ln 1.8 = 9.4 - 2 \ln 1.8 \) and \( 2x \ln 1.8 = 30 - 2 \ln 1.8 \), rearranging gives the form \( x = \frac{y - 2 \ln 1.8}{2 \ln 1.8} \), where \( y \) is 9.4 or 30.
  • Using such rearrangements lets us plug into calculable forms, leading to accurate results, like \( x \approx 2.888 \) and \( x \approx 8.794 \).
Each step in solving equations should be precise to avoid rounding errors and maintain accuracy. Remember, consistency in units and precision in the solution process is key.

Verifying solutions by back-substitution into the original equation ensures our computations are correct. This step is valuable for ensuring that derived results are reliable.

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Most popular questions from this chapter

Dog Population The number of pet dogs in the United States can be represented as \(p(t)\) million where \(t\) is the number of years since \(2003 .\) (Source: Pet Food Institute) a. Write a sentence of interpretation for \(p(0)=61.5\). b. Write the function notation for the statement "There were 66.3 million pet dogs in the United States in \(2008 .\) "

The total amount of credit card debt \(t\) years since 2010 can be expressed as \(D(t)\) billion dollars. The number of credit card holders is \(N(t)\) million cardholders, \(t\) years since 2010 (with some people having more than one credit card). a. Write an expression for the average credit card debt per cardholder. b. Write a model statement for the expression in part \(a\).

For Activities 19 through 22, each data set can be modeled using a logarithmic function. Interchange the inputs and outputs on the given tables of data and write models for the inverted data. Piroxicam Concentration in the Bloodstream $$ \begin{array}{|c|c|} \hline \text { Days } & \begin{array}{c} \text { Concentration } \\ (\mu \mathrm{g} / \mathrm{ml}) \end{array} \\ \hline 1 & 1.5 \\ \hline 3 & 3.2 \\ \hline 5 & 4.5 \\ \hline 7 & 5.5 \\ \hline 9 & 6.2 \\ \hline 11 & 6.5 \\ \hline 13 & 6.9 \\ \hline 15 & 7.3 \\ \hline 17 & 7.5 \\ \hline \end{array} $$

Write the inverse for each function. $$ s(t)=-2 t-4 $$

A company posted a net loss of \(\$ 3\) billion during the 3rd quarter. During the same quarter, the company's revenue was \(\$ 5\) billion. a. What was the company's cost in the 3 rd quarter? b. Assuming \(R(t)\) represents revenue and \(P(t)\) represents profit during the \(t\) th quarter, write an expression for cost.
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