Problem 64 - Elementary and Intermediate Algebra

Question

Solve each equation. Express all answers to four decimal places. See Example 5. $$ \ln x=0.926 $$

Step-by-Step Solution

Answer

The solution is approximately \( x = 2.5245 \).

Step 1: Understand the problem

We need to solve the equation \( \ln x = 0.926 \) for the variable \( x \). The natural logarithm, \( \ln \), is the inverse of the exponential function \( e \). The goal is to express \( x \) in direct terms.

Step 2: Isolate the Variable

Since \( \ln x = 0.926 \), we need to eliminate the \( \ln \) by applying the exponential function (\( e \)) to both sides of the equation. This step utilizes the property that \( e^{\ln x} = x \).

Step 3: Apply Exponential Function

Raise the number \( e \) to the exponent of both sides of the equation: \[ x = e^{0.926} \]This simplifies the equation by making use of the inverse nature of \( \ln \) and \( e \).

Step 4: Calculate the Exponential

Calculate the right side using a calculator to find \[ x = e^{0.926} \approx 2.5245 \]Ensure the answer is rounded to four decimal places as required.

Key Concepts

Exponential Function

The exponential function is one of the most important mathematical functions and is denoted by \( e^x \), where \( e \) is a constant approximately equal to 2.71828. It forms the basis of the natural logarithm, and it plays a crucial role in many areas of mathematics and science. Some key properties of the exponential function include:

• It is defined for all real numbers, and it is a continuous and smooth curve.
• The exponential function increases rapidly, and it is strictly positive for all real numbers.
• Its derivative and integral are unique in that \, \( \frac{d}{dx}e^x = e^x \) and \( \int e^x \, dx = e^x + C \), where \( C \) is the constant of integration.

By applying the exponential function to both sides of an equation with a logarithm, we can "undo" the logarithmic operation. This property is particularly useful when working with natural logarithms to isolate variables in an equation.

Inverse Function

An inverse function essentially reverses the operation of a given function. If you have a function \( f(x) \) and its inverse \( f^{-1}(x) \), then applying \( f \) and then \( f^{-1} \) (or vice versa) will yield the original value: \( f(f^{-1}(x)) = x \). For the natural logarithm \( \ln(x) \) and the exponential function \( e^x \), these functions are inverses of each other:

• For any positive value \( x \), \( \ln(e^x) = x \) and \( e^{\ln(x)} = x \).
• This inverse relationship allows us to solve equations involving natural logs by using exponentials and vice versa.

Understanding how these inverse relationships work allows you to apply them strategically, particularly when simplifying complex equations that involve logarithmic and exponential terms.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. Here, we focus on equations involving logarithms and exponentials, such as \( \ln x = 0.926 \). To solve such an equation:

• Identify the inverse function that can be used to isolate the variable. Since \( \ln x \) is the natural logarithm, we use the exponential function as its inverse.
• Apply the exponential function to both sides of the equation, which yields \( e^{\ln x} = e^{0.926} \). Because of the inverse relationship, this simplifies to \( x = e^{0.926} \).
• Finally, use a calculator to compute the numerical value, which gives approximately \( x = 2.5245 \).

The process of applying the inverse function and performing calculations is fundamental in solving such mathematical equations correctly.

Rounding Decimals

Rounding decimals is an important mathematical skill that ensures precision in numerical results while avoiding unnecessary complexity. When instructed to round to four decimal places:

• Identify the fourth decimal place. For example, in the number 2.524523, the fourth decimal place is 4.
• Check the digit immediately following it. If it's 5 or greater, increase the fourth decimal place by one. In our example, it's 5, so we round up to 2.5245.
• If the digit is less than 5, leave the fourth decimal place unchanged.

This technique helps when precision is required in reporting the results of calculations, ensuring consistency in data representation across different mathematical solutions.