Problem 54 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, solve for \(x\) or \(t\). $$ 2 \ln x=4 $$

Step-by-Step Solution

Answer

The solution to the equation \(2 \ln x = 4\) is \(x \approx 7.389\).

Step 1: Divide both sides of the equation by 2

Divide both sides of the equation \(2 \ln x = 4\) by 2 to isolate the \(\ln\) term. As a result, the equation transforms to \(\ln x = 2\).

Step 2: Conversion to Exponential form

Convert the equation from logarithmic form to exponential form. The rule that will be used in this situation is that \(\ln a = b\) can be rewritten as \(e^{b} = a\). Applying the rule to the simplified equation from Step 1, \(x\) becomes \(e^2\).

Step 3: Evaluate \(e^2\)

Evaluate \(e^2\) to get the value of \(x\). The Euler number \(e\) approximates to 2.71828, so \(e^2\) will be approximately \(7.389056098930649572\).

Key Concepts

Exponential Functions

Understanding exponential functions is crucial for solving logarithmic and many algebraic equations. Exponential functions involve expressions where a constant base is raised to a variable exponent. They can be written in the form of \(a^x\), where \(a\) is a constant. These functions show exponential growth or decay, depending on the base's value:

• If \(a > 1\), the function represents exponential growth.
• If \(a < 1\), the function represents exponential decay.

In the context of logarithmic equations, these functions help to reverse the logarithmic operation. For example, consider converting a logarithmic equation like \( \ln x = 2 \) to its exponential form: here, "\(e\)" is used as the base in natural logarithms, transforming the equation to \(x = e^2\). This conversion is key to understanding and solving such equations effectively, allowing you to go from a logarithm back to a straightforward number.

Natural Logarithm

The natural logarithm, denoted by \( \ln \), is a special type of logarithm with a base \(e\), where \(e\) is approximately 2.71828. It is commonly used in various applications, particularly in calculus and higher-level mathematics. The natural logarithm of a number \(y\) answers the question: "To what power must \(e\) be raised, to obtain \(y\)?"

Natural logarithms are particularly useful because their calculus-related properties make them ideal for solving certain algebraic equations. For instance, when you encounter an equation such as \( \ln x = 2 \), you can convert it back to a more familiar exponential form, \( x = e^2 \), to find the value of \(x\). This conversion is a powerful tool when working with exponential and logarithmic relationships.

• The notation \( \ln x \) is shorthand for log base \(e\), or logarithm to the natural base.
• The natural logarithm is inverse to the exponential function with base \(e\).

These features of the natural logarithm help simplify complex relationships among variables.

Solving Algebraic Equations

Solving algebraic equations involves finding the values of unknown variables that satisfy the equation. To solve an equation, you must isolate the variable on one side of the equation using inverse operations. In the example given, the original equation is \(2 \ln x = 4\). Here are the steps applied in solving it:

• First, simplify the equation by dividing both sides by 2 to get \( \ln x = 2\).
• Next, use the property of logarithms to convert \( \ln x = 2\) into its exponential form: \(x = e^2\).
• Finally, compute \(e^2\) to determine the value of \(x\), which is approximately 7.39.

By breaking down the process into these steps, you can tackle more complex equations in a structured way. Knowing these fundamental strategies can help you approach a wide range of algebraic problems with confidence.