Problem 32 - Precalculus : Building Concepts and Connections

Question

Solve the exponential equation. Round to three decimal places, when needed. $$4 e^{x}=-x+3$$

Step-by-Step Solution

Answer

The solution to the equation \(4 e^{x}=-x+3\), to three decimal places, will vary based on the starting point for the numerical approximation method used. Always use a reliable calculator or software to solve such equations.

Step 1: Isolate the exponential term on one side of the equation

First, rearrange the given equation to isolate the exponential term on one side: \(4e^x = -x + 3\)

Step 2: Apply the natural logarithm to both sides

Applying the natural logarithm (ln) to both sides can help us eliminate the exponential form and solve for the variable x: \(ln(4e^x)=ln(-x+3)\).

Step 3: Apply the property of logarithms

By the rule of logarithms, you can split the left side of equation: \(ln(4) + ln(e^x)=ln(-x+3)\). We know that \(ln(e^x)=x\). Substituting this into the equation gives \(ln(4) + x=ln(-x+3)\).

Step 4: Solve for x

Now, the equation \(x + ln(4) = ln(-x + 3)\) is nonlinear and cannot be solved for x analytically. We need to use a numerical method such as Newton's method or the bisection method to solve for x. This can be done using a graphing calculator or a computer software that can perform numerical approximations. Make sure to round to three decimal places.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as ln, is a powerful mathematical function. It is particularly useful in solving exponential equations where exponential terms need to be isolated. The natural logarithm is the inverse of the exponential function \( e^x \). This means when you apply ln to \( e^x \), you effectively invert the relationship, which helps in isolating the variable.
The base of the natural logarithm, \( e \), is approximately 2.71828, known as Euler's number. This base is special because it simplifies many equations in both calculus and logarithms. For example, the property \( ln(e^x) = x \) follows directly from this definition.
In the context of our equation \( 4e^x = -x + 3 \), using ln helps transform the equation into a more manageable form. By isolating \( e^x \) and applying ln, we can chip away at the complexity and get closer to pinpointing the value of \( x \). This transformation using ln can then be leveraged through other mathematical strategies or numerical methods when analytical solutions aren't possible.

Numerical Methods

Numerical methods are techniques used to find approximate solutions to mathematical problems that cannot be solved analytically. They are often used in solving nonlinear equations, like the one in our problem \( x + ln(4) = ln(-x + 3) \). Since this equation doesn't allow a straightforward algebraic solution, numerical methods come into play.
Common numerical methods include:

• Newton's Method: Utilizes the function's derivative to successively approximate the root.
• Bisection Method: Narrows down the interval in which the root lies by repeatedly dividing it in half and choosing subintervals.

For our equation, employing such methods would involve using graphing tools or software to determine where the function balances out (or equals zero). This approach helps in honing in on the approximate value of \( x \) to the desired precision of three decimal places. Numerical methods provide a practical solution when an equation is too complex for simple algebra alone.

Properties of Logarithms

Understanding the properties of logarithms is crucial when working with exponential equations. These properties allow us to transform and simplify complex equations, making them more convenient to solve.
Here are some essential properties used frequently:

• Product Property: \( ln(ab) = ln(a) + ln(b) \)
• Quotient Property: \( ln(a/b) = ln(a) - ln(b) \)
• Power Property: \( ln(a^b) = b \cdot ln(a) \)
• Inverse Property: \( ln(e^x) = x \)

In our exercise, we used the product and inverse properties of logarithms. Starting with the equation \( ln(4e^x) = ln(-x+3) \), the left side was expanded using the product property into \( ln(4) + ln(e^x) \). Recognizing that \( ln(e^x) \) simplifies to \( x \), the equation became \( x + ln(4) = ln(-x + 3) \). This transformation highlights how the properties of logarithms are harnessed to pave the way for numerical solutions when algebraic manipulations aren't sufficient.