Problem 30 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each exponential equation by taking the logarithm on both sides. Express the solution set in terms of logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$e^{0.08 x}=4$$

Step-by-Step Solution

Answer

The solution to the equation \( e^{0.08x} = 4 \) is approximately \( x = 13.86 \) up to two decimal places.

Step 1: Apply Logarithm

By applying natural logarithm (ln) to both sides of the equation, we have: \( \ln(e^{0.08x}) = \ln(4) \)

Step 2: Use Log Property

By the power rule of logarithms, the exponent in \( \ln(e^{0.08x}) \) can be moved in front to get: \( 0.08x \times \ln(e) = \ln(4) \). Since the logarithm of e (base e) is 1, it simplifies to \( 0.08x = \ln(4) \)

Step 3: Isolate x

To isolate x, we divide both sides of the equation by 0.08. This gives: \( x = \ln(4) / 0.08 \)

Step 4: Decimal Approximation

Finally, we use a calculator to find a decimal approximation for \( x \). This yields \( x \approx 13.86 \), correct to two decimal places.

Key Concepts

Natural Logarithms

Natural logarithms, often abbreviated as "ln," are logarithms that use the base of Euler's number, denoted as \(e\). Euler's number, \(e\), is an irrational number approximately equal to 2.71828. In mathematics, natural logarithms are used widely because they simplify a lot of mathematical expressions and calculations due to the special properties of \(e\).
In exponential equations, like the one given in the problem \(e^{0.08x}=4\), taking the natural log of both sides is a common method. It helps in bringing the exponent down where it can be easily managed. As shown in the solution, applying \(\ln\) both sides of the equation helps us convert the problem into a form where the tools of algebra, like multiplication and division, can be used to find the variable \(x\). This is because the natural log of \(e\), or \(\ln(e)\), is always 1, which greatly simplifies calculations.

Logarithm Properties

Logarithm properties are foundational tools that help us manage and simplify various complex equations. Some important properties include:

• Product Rule: \(\ln(ab) = \ln(a) + \ln(b)\)
• Quotient Rule: \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)
• Power Rule: \(\ln(a^b) = b \cdot \ln(a)\)

In the step-by-step solution, the Power Property is used to solve the equation. By applying this property, the exponent rolls down into a coefficient, allowing the equation to be more straightforward in solving. For example, \(\ln(e^{0.08x}) = 0.08x \cdot \ln(e)\), leveraging the fact that \(\ln(e) = 1\), further simplifies the equation to \(0.08x = \ln(4)\). These properties are efficient techniques for solving logarithmic and exponential equations.

Decimal Approximation

Once the solution of an equation is expressed in terms of logarithms, it's often necessary to find its decimal approximation, especially for educational or practical purposes. In this problem, after isolating \(x\) in the equation \(x = \ln(4) / 0.08\), we calculate its decimal value.
Using a calculator, input \(\ln(4)\) to get approximately 1.3863. Dividing this number by 0.08 gives \(x \approx 13.86\), rounded to two decimal places. This step is crucial for providing a comprehensible answer that can easily be used in real-world applications. Accurate decimal approximations help us understand the scale and implications of an answer without working in terms of logarithms, which might be abstract for some students.

Solving Equations

Solving equations, especially those involving exponential and logarithmic functions, often requires a series of algebraic steps designed to isolate the variable of interest. In the given problem \(e^{0.08x}=4\), the goal is to solve for \(x\).
The steps include taking the natural log of both sides to help bring the exponent down, using the properties of logarithms to simplify the terms, and then isolating the variable by mathematical operations like division or multiplication.\(x = \ln(4) / 0.08\).
Such systematic approaches are invaluable tools for tackling an array of mathematical problems, helping students develop a structured way of thinking about and solving complex equations. By understanding and following these steps, students can enhance their problem-solving skills, which are applicable in many disciplines beyond mathematics itself.