Problem 32 - Intermediate Algebra

Question

For Problems \(31-40\), use your calculator to find \(x\) when given \(\ln x\). Express answers to five significant digits. $$ \ln x=0.9413 $$

Step-by-Step Solution

Answer

\(x \approx 2.5639\)

Step 1: Understand the Problem

We need to find the value of \(x\) given that \(\ln x = 0.9413\). Here, \(\ln\) represents the natural logarithm, which is the logarithm to the base \(e\). So, we want to solve the equation \(\ln x = 0.9413\).

Step 2: Rearrange the Equation

To find \(x\), we need to convert the logarithmic equation into an exponential form. The property of natural logarithms tells us that if \(\ln a = b\), then \(a = e^b\). Thus, we have \(x = e^{0.9413}\).

Step 3: Use a Calculator to Compute

Now, we use a calculator to find the value of \(x\) by evaluating \(e^{0.9413}\). Make sure the calculator is set to the correct mode to handle natural exponentiation.

Step 4: Round the Result

Once the calculator gives the result, we take the number and round it to five significant digits to comply with the problem's requirements.

Key Concepts

Understanding Natural Logarithms

The natural logarithm, denoted as \(\ln\), is a specific type of logarithm that uses the mathematical constant \(e\) as its base. The constant \(e\) is approximately equal to 2.71828 and has special properties that appear often in calculus and complex analysis.
When you see \(\ln x\), it represents the power you must raise \(e\) to get \(x\).
For example, if \(\ln x = y\), this tells us that \(e^y = x\). Understanding this relationship is crucial for converting logarithmic equations to exponential form.

Calculator Usage for Exponential Equations

When solving exponential equations like \(x = e^{0.9413}\), using a calculator becomes essential. Modern calculators usually have a dedicated button or function for natural exponential calculations, denoted as \(e^x\).
Here's how you can use it effectively:

• Ensure your calculator is in the correct mode, often labeled as "scientific" or "math" mode, to use exponential functions.
• Enter the exponent (0.9413 in this case) directly after selecting the \(e^x\) function.
• Press "Enter" or "=" to get the result.
• Double-check your calculator input to avoid common errors, like entering extra numbers or forgetting the exponentiation mode.

These steps will help you accurately compute values involving natural exponentials.

Rounding to Significant Digits

Significant digits are the digits that carry meaningful information about a number's precision. When a problem specifies rounding to five significant digits, you need to follow specific rules:

• Identify the first non-zero digit. This is your first significant digit.
• Continue counting until you've identified five digits.
• Round the number based on the value of the digit immediately following the fifth significant digit. If this digit is 5 or greater, round up.
• If it's less than 5, leave the fifth digit as is.

Applying these rules ensures you maintain the necessary level of precision needed for scientific computation.

From Logarithmic to Exponential Form

Converting logarithmic equations to exponential form is a powerful tool in solving equations involving natural logarithms.
The conversion relies on the fundamental logarithmic identity: if \(\ln a = b\), then \(a = e^b\). This transformation allows us to isolate the variable we are solving for.
In our example, we started with the equation \(\ln x = 0.9413\). By using the property mentioned above, we convert it to the exponential form: \(x = e^{0.9413}\).
Understanding this process is essential for tackling more complex equations, as it provides a way to manipulate and solve the problem by translating it into a more familiar arithmetic form.