Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Solve exactly: \(e^{x-4}=6\)

Short answer

The solution is \(x \approx 5.79176\).

Step-by-step solution

Isolate the Exponential Expression

The given equation is \(e^{x-4} = 6\). Notice that the exponential expression \(e^{x-4}\) is already isolated.

Apply the Natural Logarithm

To solve for \(x\), apply the natural logarithm (ln) to both sides of the equation. Using the property of logarithms \( \ln(e^y) = y \): \( \ln(e^{x-4}) = \ln(6) \).

Simplify using Logarithm Properties

Use the property \( \ln(e^{x-4}) = x-4 \). So the equation becomes: \( x - 4 = \ln(6) \).

Solve for x

Add 4 to both sides to isolate \(x\): \( x = \ln(6) + 4 \).

Compute the Value

Use a calculator to find the numerical value of \( \ln(6) \): \( \ln(6) \approx 1.79176 \). Therefore, \( x \approx 1.79176 + 4 = 5.79176 \).

Key concepts

Understanding the Natural Logarithm

The natural logarithm, written as \(\text{ln}\), is a specific logarithm to the base \(\text{e}\). The number \(\text{e}\) (approximately 2.718) is an irrational constant and is very important in mathematics, especially in calculus and complex analysis. When we apply the natural logarithm to \(e^x\), we get \(\text{ln}(e^x) = x\), which is a very useful property for solving exponential equations. For example, if we have \(e^{x-4}=6\), we can apply the natural logarithm to both sides to help us isolate and solve for \(x\). This property simplifies complex exponential terms by bringing the exponent down to a basic expression.

Key Logarithm Properties

Logarithm properties are essential in simplifying and solving equations, especially ones involving exponents. Here are some important properties to remember:

• \text{\textbf{log}}_b(xy) = \text{\textbf{log}}_b(x) + \text{\textbf{log}}_b(y)\: This property states that the logarithm of a product is the sum of the logarithms.
• \text{\textbf{log}}_b\bigg(\frac{x}{y}\bigg) = \text{\textbf{log}}_b(x) - \text{\textbf{log}}_b(y)\: This property shows that the logarithm of a quotient is the difference of the logarithms.
• \text{\textbf{log}}_b(x^n) = n \text{\textbf{log}}_b(x)\: Here, the logarithm of a number raised to an exponent is the exponent times the logarithm of the base number.

Using these properties helps in breaking down more complex expressions into simpler ones, making them easier to solve. For example, in our exercise, using \( \text{ln}(e^{x-4}) = x-4 \) simplifies the equation significantly.

Step-by-Step Equation Solving

Solving equations step-by-step can feel daunting, but breaking it into smaller parts makes it manageable. Let’s solve an exponential equation as an example:

• Step 1: Isolate the Exponential Expression
  Start with the given equation \(e^{x-4}=6\). Notice that the exponential term is already isolated, simplifying our work.
• Step 2: Apply the Natural Logarithm
  Apply \( \text{ln} \) to both sides, which uses the property \( \text{ln}(e^y) = y \). This gives us \( \text{ln}(e^{x-4}) = \text{ln}(6) \).
• Step 3: Simplify using Logarithm Properties
  Use the property \( \text{ln}(e^{x-4}) = x-4 \). The equation now simplifies to: \( x-4 = \text{\textbf{ln}}(6) \).
• Step 4: Solve for \( x \)
  Add 4 to both sides to isolate \( x \): \( x = \text{ln}(6) + 4 \).
• Step 5: Compute the Value
  Use a calculator to find \( \text{ln}(6) \) which is approximately 1.79176. Therefore, \( x \) is approximately \( 1.79176 + 4 = 5.79176 \).

Understanding and practicing these steps can make solving similar equations much easier.