Problem 63 - Intermediate Algebra

Question

Solve each equation. Express all answers to four decimal places. $$ \ln x=4.24 $$

Step-by-Step Solution

Answer

The solution is \( x \approx 69.9716 \).

Step 1: Understand the Problem

We are given the equation \(ln x = 4.24\). Our goal is to solve for the value of \( x \). The equation involves the natural logarithm, \(ln\), which is the logarithm to the base \( e \).

Step 2: Apply the Inverse Function

The natural logarithm can be undone using the exponential function with base \(e \). Hence, to isolate \( x \), we apply the inverse operation: \( e^{ln x} = e^{4.24} \). This gives us \( x = e^{4.24} \).

Step 3: Calculate the Exponential Value

Using a calculator, compute the value of \( e^{4.24} \). The exponential expression corresponds to raising the number \(e\) to the power of \(4.24\).

Step 4: Round the Result

The calculation from the previous step results in \( x \approx 69.9716 \). Since we need to express the answer to four decimal places, we have \( x = 69.9716 \).

Key Concepts

Understanding the Exponential Function

The exponential function, denoted as \( e^x \), is a mathematical function that involves raising the mathematical constant \( e \) to the power of \( x \). The constant \( e \) is approximately equal to 2.71828. It's like a special number used in mathematics, especially when dealing with growth and decay processes.

• The function \( e^x \) represents how quantities grow exponentially rather than linearly.
• This means that the rate of change of the function's value increases rapidly.
• For example, we can see exponential growth in situations such as compound interest, population growth, and certain scientific phenomena.

Understanding the exponential function is crucial when dealing with inverse operations involving natural logarithms, as it's the function that effectively 'undoes' the operation of logarithms.

Inverse Operations in Mathematics

Inverse operations are mathematical operations that 'undo' each other. In simpler words, they get us back to where we started. For example, addition and subtraction are inverse operations. For logarithms and exponential functions, these inverse operations are especially important:

• The natural logarithm, \( \ln(x) \), is the inverse operation of the exponential function \( e^x \).
• This means if you take the natural logarithm of a number, then apply the exponential function, you'll return to the original number.
• So, if you have \( \ln(x) = a \), applying the exponential function leads back to \( x = e^a \).

These operations are essential in transforming equations involving logarithms into more solvable forms, making complex problems simpler to handle.

Solving Equations Involving Logarithms

When solving equations involving logarithms, like \( \ln x = 4.24 \), understanding how to manipulate and use inverse operations is key. Here's a simplified way to think about solving such equations:

• Identify the variable within the logarithm that you need to solve for.
• Apply the exponential function to both sides of the equation to eliminate the logarithm and isolate the variable.
• Use a calculator to compute the value of the exponential expression accurately.

In our example, we transformed \( \ln x = 4.24 \) by applying the inverse operation, yielding \( x = e^{4.24} \). Using a calculator gives the numeric result, \( x \approx 69.9716 \), rounding it to four decimal places as the final step. Following these steps not only solves the equation but also reinforces understanding of the interplay between logarithms and exponential functions.