Problem 63 - Elementary and Intermediate Algebra

Question

Solve each equation. Express all answers to four decimal places. See Example 5. $$ \ln x=4.24 $$

Step-by-Step Solution

Answer

\( x = 69.0070 \)

Step 1: Understanding the Natural Logarithm

The equation given is \( ln x = 4.24 \). The symbol \( ln \) refers to the natural logarithm, which means it's logarithm with base \( e \), where \( e \approx 2.71828 \). The equation states that \( e \) raised to some exponent will equal \( x \).

Step 2: Convert to Exponential Form

To solve \( ln x = 4.24 \), we need to convert the logarithmic equation into its equivalent exponential form. The equivalent form is \( x = e^{4.24} \).

Step 3: Calculating the Exponent

Now, we calculate \( e^{4.24} \). Using a calculator, compute this expression to find \( x \).

Step 4: Result Rounding

After calculating \( e^{4.24} \), ensure the result is rounded to four decimal places to meet the requirements of the problem. Thus, \( x = 69.0070 \).

Key Concepts

Exponential Equations

Exponential equations are equations in which variables appear as exponents. These types of equations are prevalent in various fields, including science and engineering, as they help model growth processes, like population growth and radioactive decay.
When dealing with exponential equations, the goal is generally to isolate the variable of interest.

• The natural logarithm (\(ln\)) can be used to solve exponential equations where the base is \(e\), a constant approximately equal to 2.71828.
• By applying logarithmic principles, you can convert equations into more manageable algebraic forms.

In the exercise provided, the use of natural logarithms transforms a logarithmic equation into an exponential one, making it easier to solve for \(x\). Understanding this relationship between logarithmic and exponential forms is key to solving many such equations.

Exponential Form

The exponential form of an equation is a way of expressing equations where a particular number is raised to a power or exponent. Exponential form is invaluable when dealing with equations involving growth or decay, such as compound interest or biological processes.

• In the context of natural logarithms, the exponential form involves expressing \(x\) as a power of \(e\).
• In the solved example, converting \(\ln x = 4.24\) to \(x = e^{4.24}\) is a crucial step in solving the equation.

This process of converting logarithmic expressions to exponential form often simplifies calculations, allowing the use of calculators or computational tools to find precise numerical solutions.

E Constant

The constant \(e\) is one of the most important numbers in mathematics, often referred to as Euler's number. It is the base of the natural logarithm and is approximately equal to 2.71828.

• \(e\) is central in defining the exponential function \(e^x\), which describes continuous growth.
• The unique property of the function \(e^x\) is that its rate of growth is proportional to its current size at any given point on its curve.

In mathematics and its numerous applications like calculus and complex analysis, \(e\) provides a way to relate exponential growth scenarios to logarithmic scales. Understanding \(e\) and its applications is fundamental to mastering these subjects, including the transformation from logarithmic to exponential expressions as seen in the given exercise.