Problem 41 - College Algebra and Calculus: An Applied Approach

Question

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(e^{3 x}=12\)

Step-by-Step Solution

Answer

The solution to the equation \(e^{3x} = 12\) is \(x \approx 0.792\).

Step 1: Write down the equation

The equation we need to solve is \(e^{3x} = 12\).

Step 2: Remove the exponent

To remove the exponent on \(e\), take the natural logarithm on both sides of the equation: \(\ln(e^{3x}) = \ln(12)\) . Now, apply the natural logarithm property that allows us to move the exponent on \(e\) in front: \(3x \times \ln(e) = \ln(12)\) . Remember, \(\ln(e) = 1\), so our equation now simplifies to \(3x = \ln(12)\).

Step 3: Solve for x

To solve for \(x\), divide both sides of the equation by 3: \(x = \frac{\ln(12)}{3}\).

Step 4: Approximate the result

To get the value for \(x\), approximate \(\frac{\ln(12)}{3}\) to three decimal places using a calculator. \(x \approx 0.792\).

Key Concepts

Natural Logarithm

The natural logarithm, commonly denoted as \( \ln \), is a logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. The natural logarithm is used frequently in calculus and mathematical analysis because it can simplify the process of dealing with exponential growth or decay.

• It is the inverse function of the exponential function \( e^x \). This means that if \( y = \ln(x) \), then \( x = e^y \).
• This inverse property is particularly useful because when you apply \( \ln \) to both sides of an equation with an exponential function, the exponent can be brought down. For example, \( \ln(e^{3x}) = 3x \cdot \ln(e) = 3x \).
• One special property of the natural logarithm is that \( \ln(e) = 1 \), which often simplifies calculations significantly.

Using these properties, we can convert an exponential equation into a linear form, which is usually much easier to solve.

Solving Algebraic Equations

Solving algebraic equations involves finding the value(s) of variable(s) that make the equation true. The problem outlined here is an exponential equation, which requires specific techniques.In our case, the equation \( e^{3x} = 12 \) involves solving for \( x \), the unknown variable. By applying the natural logarithm, we simplify the problem:

• First, take \( \ln \) on both sides, which gives \( \ln(e^{3x}) = \ln(12) \).
• Using the property that allows us to move the exponent, we get \( 3x = \ln(12) \).
• This transforms the exponential equation into a linear one, \( 3x = \ln(12) \), that can now be solved using basic algebraic operations.

To isolate \( x \), divide both sides by 3, and compute using a calculator: \( x = \frac{\ln(12)}{3} \). Approximating gives the solution as \( x \approx 0.792 \) after rounding to three decimal places.

Exponential Functions

Exponential functions form a fundamental part of mathematics, characterized by their constant base raised to a variable exponent. The formula is usually expressed as \( a^x \), where \( a \) is a positive constant. In the context of this equation, \( e^{3x} \) is an entity where \( e \) is the base and \( 3x \) is the exponent.

• Exponential functions are known for rapid growth or decay, depending on the base \( a \) and the sign of the exponent \( x \).
• If \( a > 1 \), the function grows with increasing \( x \). If \( 0 < a < 1 \), it decays.
• When dealing with \( e \), we’re usually interested in continuous growth contexts, such as population growth or radioactive decay.

Understanding the nature of exponential functions is crucial for effectively handling and solving problems involving them, like the equation \( e^{3x} = 12 \). This allows us to utilize logarithms to simplify and solve for the unknown variable.