Problem 59 - College Algebra with Modeling and Visualization

Question

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$e^{-x}=3$$

Step-by-Step Solution

Answer

The solution is approximately \( x = -1.10 \).

Step 1: Isolate the Exponential Expression

The given equation is \( e^{-x} = 3 \). Here, we need to isolate \( x \) by taking the natural logarithm of both sides to remove the exponential. This step will allow us to work with the base \( e \), which simplifies our calculations.

Step 2: Apply the Natural Logarithm

Apply the natural logarithm to both sides of the equation: \( \ln(e^{-x}) = \ln(3) \). Using the properties of logarithms, \( \ln(e^{-x}) \) simplifies to \(-x \cdot \ln(e) = -x \) because \( \ln(e) = 1 \). Thus, the equation becomes \(-x = \ln(3) \).

Step 3: Solve for x

To solve for \( x \), divide both sides of the equation by \(-1\): \( x = -\ln(3) \). Use a calculator to approximate \( \ln(3) \).

Step 4: Approximate the Solution

Use a calculator to find \( \ln(3) \approx 1.10 \). Then \( x = -1.10 \). So, the solution is \( x \approx -1.10 \) to the nearest hundredth.

Key Concepts

Change of Base Formula

The Change of Base Formula is a fundamental tool when dealing with logarithms, especially when you don't have a calculator that can handle arbitrary bases. It allows us to express a logarithm in terms of logs of any base of our choice, usually either base 10 or base e (natural logarithm), which are typically supported by calculators. The formula states:

• \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \)

Here:

• \( b \) is the base you want to change from.
• \( a \) is the number you are taking the log of.
• \( c \) is the new base you want to convert to; often this is 10 or e.

By using this formula, you can solve logarithms of any base by converting them into terms of base 10 or the natural logarithm. This is handy while solving equations that initially seem more complex than they are.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a specific logarithm with the base e. The number e is approximately \( 2.71828 \), and it's an important constant in mathematics, especially within calculus and exponential functions. The natural logarithm possesses several key properties:

• It is the inverse of the exponential function \( e^x \).
• \( \ln(e) = 1 \) because the natural log of e to the power of 1 is just 1.
• \( \ln(1) = 0 \) because any number to the power of 0 is 1.
• It simplifies the process of solving exponential equations like \( e^{-x} = 3 \).

In this way, the natural logarithm simplifies calculations with exponential equations, allowing us to easily solve for variables.

Properties of Logarithms

Understanding the properties of logarithms is crucial for solving equations involving logarithms and exponents. These properties transform complex expressions into simpler forms, aiding in the solution process. Key properties include:

• Product Rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \)
• Quotient Rule: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \)
• Power Rule: \( \log_b(m^n) = n\log_b(m) \)
• Change of Base Rule: \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \)

Using these rules, expressions that look intimidating at first glance can be broken down into simpler parts. For example, while solving \( \ln(e^{-x}) \), the Power Rule helps us simplify it by bringing \(-x\) in front, resulting in \(-x \cdot \ln(e)\), which further reduces to \(-x\).

Solving Equations

Solving equations involving logarithms and exponentials often involves a series of strategic steps to simplify the problem. The primary goal is typically to isolate the variable in question, using logarithmic rules to remove or simplify logarithmic expressions. Here’s a general approach:

• Isolate the Exponential Part: Try to get the exponential expression by itself if possible, just like in the example \( e^{-x} = 3 \).
• Apply Logarithms: Use logarithms to remove the exponential base. This could mean using the natural log \( \ln \) when working with base e or the common log \( \log \) when using base 10.
• Simplify Using Properties: Deploy properties of logarithms to simplify the expression further, making it easier to solve for the variable.
• Precise Approximation: Once the equation is simplified, a calculator can approximate any irrational numbers to a specific decimal place, like the nearest hundredth. For example, \( \ln(3) \approx 1.10 \) leads to \( x \approx -1.10 \).

By following these steps systematically, you can solve complex equations for unknown variables with ease, even when involving exponential or logarithmic expressions.