/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 Solve the following for \(r\). ... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 22

Solve the following for \(r\). a. \(0.9=e^{r}\) b. \(2=e^{r}\) c. \(0.75=e^{r}\)

Short Answer

Expert verified
a. r 鈮� -0.1054 b. r 鈮� 0.6931 c. r 鈮� -0.2877

Step by step solution

01

Identify the given equation

The given equations are: a. 0.9 = e^r b. 2 = e^r c. 0.75 = e^r
02

Take the natural logarithm on both sides of each equation

To isolate the variable r, take the natural logarithm (ln) of both sides of the equation. a. ln(0.9) = ln(e^r) b. ln(2) = ln(e^r) c. ln(0.75) = ln(e^r)
03

Apply the property of logarithms

Use the property of logarithms that states ln(e^r) = r. a. ln(0.9) = r b. ln(2) = r c. ln(0.75) = r
04

Solve for r

Calculate the natural logarithm for each case to find the value of r. a. r = ln(0.9) b. r = ln(2) c. r = ln(0.75) Using a calculator: a. r 鈮� -0.1054 b. r 鈮� 0.6931 c. r 鈮� -0.2877

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Key Concepts

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

Natural Logarithm
A natural logarithm is a logarithm to the base of the mathematical constant \( e \), where \( e \) is approximately equal to 2.71828. The natural logarithm of a number x is typically denoted as \( \ln(x) \). Natural logarithms are particularly useful in calculus and solving exponential equations.
For example, in our exercise, we have expressions like \( e^r \), where we apply the natural logarithm to both sides to help solve for the variable \( r \).
Logarithmic Properties
Logarithmic properties simplify complex mathematical problems involving exponents and roots. Key properties include:
  • Product Property: \( \ln(ab) = \ln(a) + \ln(b) \)
  • Quotient Property: \( \ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b) \)
  • Power Property: \( \ln(a^b) = b \ln(a) \)
  • Inverse Property: \( \ln(e^x) = x \)
In solving the given equations, we utilized the inverse property, specifically \( \ln(e^r) = r \), to isolate \( r \).
Isolation of Variable
Isolation of a variable means rearranging an equation to express one variable explicitly in terms of the other variables. When dealing with exponential equations, we often use logarithmic functions to isolate the variable that's in the exponent.
In the exercise, starting with \(0.9 = e^r\), we took the natural logarithm of both sides to get \( \ln(0.9) = \ln(e^r) \). Using logarithmic properties, this simplifies to \( \ln(0.9) = r \), thus isolating \( r \).
Exponential Functions
Exponential functions have the form \( f(x) = a \cdot e^{bx} \) where \( e \) is Euler's number. These functions grow rapidly and are used to model a variety of real-world situations. Exponential equations often require logarithms for solving.
The given exercise involves simple exponential equations of the form \( y = e^r \), which we solve by applying the natural logarithm. This transforms the exponential form into a linear one, making it easier to find the value of \( r \).

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Most popular questions from this chapter

Expand each logarithm using only the numbers \(2,3, \log 2,\) and \(\log 3\). a. \(\log 9\) b. \(\log 18\) c. \(\log 54\)

If you drop a rubber ball on a hard, level surface, it will usually bounce repeatedly. (See the accompanying graph at the top of the next column.) Each time it bounces, it rebounds to a height that is a percentage of the previous height. This percentage is called the rebound height. a. Assume you drop the ball from a height of 5 feet and that the rebound height is \(60 \%\). Construct a table of values that shows the rebound height for the first four bounces. b. Construct a function to model the ball's rebound height, \(H,\) on the \(n\) th bounce. c. How many bounces would it take for the ball's rebound height to be 1 foot or less? d. Construct a general function that would model a ball's rebound height \(H\) on the \(n\) th bounce. where \(H_{0}\) is the initial height of the ball and \(r\) is the ball's rebound height.

Identify each of the following functions as representing growth or decay: a. \(Q=N e^{-0.029 t}\) c. \(f(t)=375 e^{0.055 t}\) b. \(h(r)=100(0.87)^{r}\)

Determine the rule(s) of logarithms that were used to expand each expression. a. \(\ln 15=\ln 3+\ln 5\) b. \(\ln 15=\ln 30-\ln 2\) c. \(\ln 49=2 \ln 7\) d. \(\ln 25 z^{3}=2 \ln 5+3 \ln z\) e. \(\ln 5 x^{4}=\ln 5+4 \ln x\) f. \(\ln \left(\frac{125}{3 x}\right)=3 \ln 5-(\ln 3+\ln x)\)

Use rules of logarithms to contract to a single logarithm. Use a calculator to verify your answer. a. \(2 \ln 3+4 \ln 2\) c. \(2(\ln 4-\ln 3)\) b. \(3 \ln 7-5 \ln 3\) d. \(-4 \ln 3+\ln 3\)
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