Problem 64 - Intermediate Algebra

Question

Solve each equation. Express all answers to four decimal places. $$ \ln x=0.926 $$

Step-by-Step Solution

Answer

The solution is approximately \( x = 2.5246 \).

Step 1: Understanding the Equation

The given equation is \( \ln x = 0.926 \). Here, \( \ln \) denotes the natural logarithm, which is the logarithm to the base \( e \), where \( e \approx 2.71828 \). The goal is to find the value of \( x \) that satisfies this equation.

Step 2: Convert the Logarithm to Exponential Form

The equation \( \ln x = 0.926 \) can be rewritten in exponential form as \( x = e^{0.926} \). This transformation uses the property that if \( \ln a = b \), then \( a = e^b \).

Step 3: Calculate the Exponential Expression

To find the value of \( x \), calculate the expression \( e^{0.926} \). Using a calculator, this results in approximately \( 2.5246 \).

Step 4: Round to Four Decimal Places

The solution needs to be expressed to four decimal places. Our calculated value \( 2.5246 \) is already in four decimal places, so this is our final answer for \( x \).

Key Concepts

Exponential Form Conversion

When you encounter a natural logarithm equation, such as \( \ln x = 0.926 \), a good first step is to convert it into its exponential form. This technique is essential because it allows us to work with the base \( e \), known as Euler's number. The natural logarithm \( \ln \) is specifically the logarithm to this base.To perform this conversion, we use the property: if \( \ln a = b \), then \( a = e^b \). This shows the direct relationship between logarithms and exponents. Hence, we can rewrite our original equation \( \ln x = 0.926 \) as \( x = e^{0.926} \).

• Remember, converting logarithmic form to exponential form is simply about rearranging the equation to solve for the unknown - in this case, \( x \).
• This technique simplifies computations and helps in solving logarithmic equations effectively.

In short, exponential form conversion is a powerful tool for understanding and solving logarithmic equations by expressing them as exponential expressions.

Euler's Number

Euler's number (\( e \)) is a fundamental constant in mathematics, approximately equal to \( 2.71828 \). It is the base of the natural logarithm, which makes it incredibly important in various fields such as calculus, financial calculations, and in solving differential equations.Understanding Euler's number is crucial when dealing with natural logarithms. It serves as the base used in exponential equations arising from converting natural logarithms. For instance, when converting \( \ln x = 0.926 \) to exponential form, \( e \) becomes the base in the expression \( x = e^{0.926} \).

• Euler's number is used because it has unique properties, like the derivative of \( e^x \) is \( e^x \), which simplifies calculus operations.
• When solving equations involving \( \ln \), \( e \) will always appear in the exponential form, bridging logarithmic and exponential representations.

Grasping Euler's number provides us with a significant advantage in comprehending and solving natural logarithm equations, making it an indispensable tool for any math enthusiast.

Solving Logarithmic Equations

To solve logarithmic equations like \( \ln x = 0.926 \), understanding the conversion to exponential form and properties of Euler's number is crucial. With the conversion from logarithmic to exponential form, we find \( x = e^{0.926} \).Here's how you proceed:

• First, express the logarithmic equation in its exponential form. This makes use of the property that \( \ln a = b \) can be rewritten as \( a = e^b \).
• Compute the exponential expression using a calculator: calculate the value of \( e^{0.926} \).
• For this problem, the calculation yields approximately \( 2.5246 \).

Solving logarithmic equations often ends with a numerical approximation, as most real-world values of \( e \) are not neatly calculable without computational tools. In mathematical problems, it's typically required to round these values to a certain number of decimal places. Here, rounding to four decimal places gives us a precise solution, \( x \approx 2.5246 \). Always ensure to follow the specifications of decimal accuracy given in the problem.