Problem 114 - A Graphical Approach to College Algebra

Question

Use any method (analytic or graphical) to solve each equation. $$e^{x+\ln 3}=4 e^{x}$$

Step-by-Step Solution

Answer

\(x = \ln \left(\frac{4}{3}\right) \approx 0.2877\)

Step 1: Simplify the Exponential Equation

Start with the given equation: \(e^{x + \ln 3} = 4e^x\). We can rewrite \(e^{x + \ln 3}\) as \(e^x \cdot e^{\ln 3}\) using the properties of exponents. Since \(e^{\ln 3} = 3\), the equation becomes \(3e^x = 4e^x\).

Step 2: Isolate and Solve for x

Now, we have \(3e^x = 4e^x\). Subtract \(3e^x\) from both sides to get \(0 = 4e^x - 3e^x\). Simplify to find \(0 = e^x\). Divide both sides by \(e^x\), giving you \(1 = \left(\frac{4}{3}\right)\).

Step 3: Take the Natural Logarithm

Since the left-hand side has been simplified to \(e^x = \frac{4}{3}\), take the natural logarithm of both sides: \(x = \ln \left(\frac{4}{3}\right)\).

Step 4: Calculate the Solution

Compute \(x = \ln \left(\frac{4}{3}\right)\) to find the numerical value of \(x\). Using a calculator, you find \(x \approx 0.2877\).

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial when solving exponential equations. These properties help us manipulate expressions in exponential form. In the given exercise, the key property of exponents used is: \(e^{a+b} = e^a \cdot e^b\). This separation into factors allows us to simplify complex expressions.
In the problem, we start with \(e^{x + \ln 3}\). By breaking it down using the property, we have \(e^x \cdot e^{\ln 3}\). Another essential exponent property to remember is that \(e^{\ln k} = k\), which simplifies \(e^{\ln 3}\) to 3. This simplification transforms the original equation into a more manageable form: \(3e^x = 4e^x\).
Recognizing and applying these properties of exponents helps reduce equation complexity and enables us to solve them systematically.

Natural Logarithm

The natural logarithm is a powerful tool in solving exponential equations, particularly when isolating variables. The natural logarithm, often expressed as \( \ln \), is the inverse of the exponential function with base \( e \). This relationship means if \( e^x = a \), then \( x = \ln a \).
In the step-by-step solution provided, once the equation is simplified to \( e^x = \frac{4}{3} \), taking the natural logarithm of both sides is an effective way to solve for \( x \).

• Apply \( \ln \) to both sides: \( \ln(e^x) = \ln\left(\frac{4}{3}\right) \).
• This uses the property that \( \ln(e^x) = x \), so \( x = \ln\left(\frac{4}{3}\right) \).

Using \( \ln \) allows us to convert from an exponential equation to a linear expression, making calculations more straightforward. Understanding the role of natural logarithms is vital in solving equations involving exponential functions.

Equation Solving Methods

When solving equations, especially those involving exponents, employing systematic methods is key. There are multiple approaches, including analytic and graphical methods. In analytic methods, algebraic manipulation—such as using the properties of exponents and logarithms—is central.
In the given exercise, the equation was simplified analytically by using properties of exponents and natural logarithms. This step-wise refinement led to a straightforward determination of \( x \).

• Initially, transform the equation using exponent properties.
• Isolate the variable by factoring out common terms.
• Apply natural logarithms to eliminate exponential terms.

Graphical methods, though not used here, involve plotting the functions to find intersection points, providing a visual insight. Both methods have their merits, but understanding and applying the correct steps in analytic methods offered a direct solution path—\( x = \ln\left(\frac{4}{3}\right) \)—which calculus or algebra often confirms. Developing a solid grasp of these methods ensures flexible problem-solving skills.