Problem 85
Question
Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ e^{3 x}=9 $$
Step-by-Step Solution
Verified Answer
Exact solution: \( x = \frac{\ln(9)}{3} \); Approximate solution to four decimal places: \( x \approx 0.7324 \).
1Step 1: Take the Natural Logarithm of Both Sides
To solve the equation \( e^{3x} = 9 \), we start by taking the natural logarithm on both sides to eliminate the exponential function. This gives us \( \ln(e^{3x}) = \ln(9) \).
2Step 2: Simplify Using Logarithmic Identities
Using the logarithmic identity \( \ln(e^a) = a \), we simplify the left side: \( 3x = \ln(9) \).
3Step 3: Solve for x
To isolate \( x \), divide both sides by 3: \( x = \frac{\ln(9)}{3} \). This is the exact solution.
4Step 4: Approximate the Result
Calculate \( \ln(9) \) using a calculator, which is approximately \( 2.1972 \). Therefore, \( x \approx \frac{2.1972}{3} \approx 0.7324 \).
Key Concepts
Understanding the Natural LogarithmExploring Logarithmic IdentitiesFinding Approximate Solutions
Understanding the Natural Logarithm
Natural logarithms are fundamental in mathematics, especially when working with exponential functions like the one in the equation \(e^{3x} = 9\). The natural logarithm, denoted as \(\ln\), is the logarithm to the base \(e\), where \(e\) is approximately 2.71828. This particular logarithm is useful because it allows us to easily handle exponential growth.
When we apply the natural logarithm to both sides of an equation, such as \(\ln(e^{3x}) = \ln(9)\), it helps remove the exponent based on the property that \(\ln(e^a) = a\). This step is crucial for solving exponential equations, as it simplifies the equation to a form that can be further manipulated to find a solution.
When we apply the natural logarithm to both sides of an equation, such as \(\ln(e^{3x}) = \ln(9)\), it helps remove the exponent based on the property that \(\ln(e^a) = a\). This step is crucial for solving exponential equations, as it simplifies the equation to a form that can be further manipulated to find a solution.
Exploring Logarithmic Identities
Logarithmic identities are formulas that simplify the process of solving logarithmic equations. In the given problem, the key identity used is \(\ln(e^a) = a\). This identity indicates that taking the natural logarithm of \(e\) raised to any power \(a\) simply returns \(a\).
This particular identity leads directly to the simplified equation \(3x = \ln(9)\). Another important logarithmic property is that logarithms turn multiplications into additions, divisions into subtractions, and powers into multiples. While other identities might not directly apply to this exercise, they can be extremely useful when dealing with more complex logarithmic and exponential equations. Understanding these identities can enhance your ability to solve various equations efficiently.
This particular identity leads directly to the simplified equation \(3x = \ln(9)\). Another important logarithmic property is that logarithms turn multiplications into additions, divisions into subtractions, and powers into multiples. While other identities might not directly apply to this exercise, they can be extremely useful when dealing with more complex logarithmic and exponential equations. Understanding these identities can enhance your ability to solve various equations efficiently.
Finding Approximate Solutions
Sometimes, after finding the exact solution to an equation, it's necessary to determine an approximate numerical value for practical use. This is especially true when dealing with irrational numbers like \(\ln(9)\), which don’t resolve neatly into a simple decimal.
To find the approximation, we calculate \(\ln(9)\) using a calculator, which gives us approximately \(2.1972\). The next step involves substituting this approximate value back into the equation \(x = \frac{\ln(9)}{3}\). Performing the division \(\frac{2.1972}{3}\) yields an approximate solution for \(x\), which is about \(0.7324\). Calculating approximate solutions is often used in real-world scenarios where exact values are either impractical or impossible to determine.
To find the approximation, we calculate \(\ln(9)\) using a calculator, which gives us approximately \(2.1972\). The next step involves substituting this approximate value back into the equation \(x = \frac{\ln(9)}{3}\). Performing the division \(\frac{2.1972}{3}\) yields an approximate solution for \(x\), which is about \(0.7324\). Calculating approximate solutions is often used in real-world scenarios where exact values are either impractical or impossible to determine.
Other exercises in this chapter
Problem 85
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Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ e^{4 x}=60 $$
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