Problem 68 - Elementary and Intermediate Algebra

Question

Solve each equation. Express all answers to four decimal places. See Example 5. $$ \ln x=-0.001 $$

Step-by-Step Solution

Answer

The solution is \( x \approx 0.9990 \).

Step 1: Understand the Problem

We have the natural logarithm equation given as \( \ln x = -0.001 \). The objective is to solve for \( x \).

Step 2: Use Exponential Form to Solve

We know that the natural logarithm \( \ln x \) is the same as \( \log_e x \). To solve for \( x \), we need to convert this equation into an exponential form. The equation \( \ln x = y \) can be rewritten as \( x = e^y \). Thus, we have \( x = e^{-0.001} \).

Step 3: Calculate the Result

To find the value of \( x \), we calculate \( e^{-0.001} \). Using a calculator or computer, compute the value: \[ x \approx e^{-0.001} \approx 0.9990 \].

Step 4: Round the Solution Appropriately

Since the problem requires rounding to four decimal places, ensure your answer is displayed to four decimal positions. In this case, \( x = 0.9990 \) is already rounded to four decimal places.

Key Concepts

Understanding Exponential Equations

Exponential equations are equations where the variable is in an exponent. In our example, the natural logarithm equation \( \ln x = -0.001 \) is solved by converting it to its exponential form. This involves understanding that the natural logarithm \( \ln x \) is the logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. Therefore, any equation of the form \( \ln x = y \) can be rewritten as an exponential equation: \( x = e^y \). When we express \( x \) in terms of \( e \), it allows us to compute the value directly using standard calculators or programming tools. This conversion to an exponential form is crucial in many areas of mathematics and science because it simplifies the solution process for logarithmic equations. It transforms the problem into one that deals with base \( e \), a fundamental constant that naturally occurs in numerous growth and decay processes, such as population growth, radioactive decay, and interest calculations.

Rounding Decimal Places

Rounding numbers is an essential skill, especially when providing answers that are both precise and easy to understand. When asked to round to a certain number of decimal places, such as four in our exercise, we look at the number immediately after our desired decimal position.

• If this number is 5 or greater, we round up the last desired decimal.
• If it's less than 5, we leave the last desired decimal as it is.

In our problem, the computation gives us \( x \approx 0.999000499833375 \ldots \), and rounding this to four decimal places results in \( x = 0.9990 \). This ensures we maintain a standard level of precision, which is particularly important in scientific and statistical analyses. Rounding helps us provide concise results, facilitating clearer communication and understanding when dealing with extensive datasets or experimental results.

Solving Equations: A Step-by-Step Approach

Solving equations involves finding the value of the variable that makes the equation true. For logarithmic equations like \( \ln x = -0.001 \), solving involves converting the equation to a more straightforward form using algebraic manipulation. Here's a systematic approach:1. **Identify the form of the equation:** Check if it's logarithmic, exponential, or any other form.2. **Isolate the variable:** Since \( \ln x \) implies \( x \) is the value plugged into the natural log, use transformations to express it as \( x = e^{-0.001} \).3. **Perform calculations:** Compute the resultant expression using a scientific calculator to find the numeric value of \( x \).4. **Round appropriately:** As discussed, ensure your answer meets the required decimal precision.Apart from accuracy, each solving step helps develop a deeper understanding of the problem, teaching you to approach mathematical problems methodically. This systematic process isn't just for math; it empowers problem-solving across various disciplines, sharpening analytical skills foundational to both academic and real-world challenges.