Problem 137 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$\frac{1+\ln x}{2}=0$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0.368\)

Step 1: Simplify the Equation

To solve this equation, start by multiplying both sides by 2 to get rid of the denominator: \(1 + \ln x = 0\)

Step 2: Isolate the natural logarithm term

Subtract 1 from both sides of the equation to isolate the natural logarithm term: \(\ln x = -1\)

Step 3: Solve for x by taking exponent

The inverse operation of the natural logarithm is the exponentiation via e (Euler's number). Therefore, we can raise both sides of the equation as a power of e to cancel out ln and solve for x: \(x = e^{-1}\)

Step 4: Calculate the numerical value

The numerical equivalent of \(e^{-1}\) is approximately 0.368. It is important to note the instruction to round the result, therefore the final answer will be obtained by rounding this value to three decimal places: \(x = 0.368\).

Key Concepts

Natural Logarithm

A natural logarithm, denoted as \( \ln \), is a logarithm whose base is the number \( e \) (approximately 2.71828), a fundamental constant in mathematics known as Euler's number. The natural logarithm is the inverse operation of exponentiation with base \( e \). It is used to transform multiplicative relationships into additive ones, which can simplify the analysis and solution of equations. For instance, in the exercise, you encounter the equation \( \frac{1+\ln x}{2}=0 \). After simplification, it turns into \( \ln x = -1 \). This means you are seeking a number \( x \) such that when the exponential function \( e^x \) is raised to \(-1\), it equals \( x \).

Exponentiation

Exponentiation is a mathematical operation involving numbers called the base and the exponent. For natural logarithms, when you solve \( \ln x = -1\), the inverse operation is to exponentiate both sides using the base \( e \). This process becomes \( x = e^{-1} \), which converts the logarithmic equation back into its original form. Thus, you are effectively asking: to what power should \( e \) be raised to get \( x \) which equals \( e^{-1} \)? This is equivalent to dividing 1 by \( e \), and the solution calculated will be \( x \approx 0.368 \). Understanding this step is critical for solving logarithmic equations using exponentiation.

Graphing Utility

A graphing utility refers to any tool, such as a graphing calculator or software, used for plotting and visualizing mathematical equations and their solutions. These tools can provide a graphical representation that helps verify results obtained algebraically. When using a graphing utility for the provided exercise:

• Input the equation \( \frac{1 + \ln x}{2} \) set equal to zero.
• Observe where the function crosses the x-axis, indicating the solution.
• In this instance, the graph should intersect at approximately \( x = 0.368 \), confirming the solution found algebraically.

Visualizing with a graphing utility reinforces understanding and provides a visual check of your calculated result.

Rounding Numbers

Rounding numbers in mathematics implies adjusting the number to a nearby and commonly-used figure—for example, something that is easier to use or approximate within acceptable error margins. In the context of this exercise, the instruction to round the result to three decimal places means once you compute \( e^{-1} \), which is approximately 0.367879, you adjust this to 0.368. Rounding helps to simplify the number without requiring extreme precision, especially when the context allows for a slight margin of error. To ensure accuracy, follow common rounding rules like:

• If the digit after the third decimal place is 5 or more, round up.
• If it's less than 5, round down.

This habit of rounding can greatly simplify calculations and results, making them more practical for use.