Problem 11 - Calculus of a Single Variable

Question

Solving an Exponential or Logarithmic Equation In Exercises 1-16, solve for \(x\) accurate to three decimal places. $$ \ln x=2 $$

Step-by-Step Solution

Answer

The solution to the equation \(\ln x = 2\) accurate to three decimal places is \(7.389\).

Step 1: Rewrite the logarithmic equation to exponential form

To solve the given logarithmic equation \(\ln x = 2\), rewrite it into exponential form. Since \(\ln\) is a natural logarithm, it is based on the base \(e\). Therefore, the logarithmic equation can be rewritten as \(e^2 = x\).

Step 2: Calculate the value of \(x\)

The next step is to calculate the value of \(x\). Since \(x = e^2\), we can find the value of \(x\) by taking \(\e^2\) with a calculator.

Step 3: Round to three decimal places

Finally, round the solution to three decimal places. The precision of the result should be within three decimal places.

Key Concepts

Natural Logarithms

Natural logarithms are a specific type of logarithm where the base is the mathematical constant \(e\), which is approximately equal to 2.71828. They are denoted by \(\ln\). The natural log is useful in many areas of mathematics and science, including growth processes, compounding interest, and in some differential equations.

• The notation \(\ln x = y\) means that \(e^y = x\). This is converting the logarithmic statement into an exponential one.
• This special base arises naturally in many mathematical contexts, particularly in calculus.

Understanding natural logarithms is crucial when dealing with exponential growth and decay, and they simplify many logarithmic expressions because the natural log of \(e\) is 1, simplifying calculations involving \(e\).
For example, solving \(\ln x = 2\) means finding the power of \(e\) that gives \(x\), leading us to \(x = e^2\).
Recognizing that \(\ln\) refers to the natural logarithm sets the stage for successfully manipulating exponential expressions.

Exponential Equations

Exponential equations are equations where variables appear as exponents. They are prevalent in many real-world applications, such as population growth, radioactive decay, and financial calculations.
Consider the equation derived from the given problem: \(e^2 = x\). Here, \(e\) is the base of the natural logarithm, and 2 is the exponent, which indicates how many times we multiply \(e\) by itself to find \(x\).

• Exponential equations can often be solved by rewriting logarithmic equations to exponential form, leveraging the properties of logarithms and exponents.
• Solutions to these equations can be easily approximated using calculators or software to handle more complex forms.

For the equation \(e^2 = x\), solving it means determining the precise value of \(x\) using the exponent 2. It involves a straightforward calculation but underscores the relationship between exponential functions and their inverses, logarithms.
This illustrates how exponential relationships can be transformed into linear ones, simplifying some mathematical processes.

Rounding Decimals

Rounding decimals is a fundamental skill in mathematics that ensures solutions are expressed with the required precision. It involves approximating a number to a specified degree of accuracy.
Here, we need to round \(e^2\) to three decimal places. Calculators and software can evaluate \(e^2\), which is approximately 7.38905609893065.

• To round to three decimal places, look at the fourth decimal: if it’s 5 or greater, round the third decimal up, otherwise keep it the same.
• This means that for \(e^2 \approx 7.38905609893065\), we round it to 7.389.

Rounding helps simplify numbers for practical use, making them easier to read and communicate. It’s especially important in science and engineering, where precision is crucial but excessive detail can be unmanageable or irrelevant.
This ensures results are both accurate and practical, adhering to specified precision guidelines.