Problem 52 - Beginning and Intermediate Algebra

Question

Solve the logarithmic equation algebraically. Then check using a graphing calculator. $$\ln (x+1)-\ln x=\ln 4$$

Step-by-Step Solution

Answer

The solution to the logarithmic equation \(\ln(x+1) - \ln x = \ln 4\) can be found by simplifying the equation using the properties of logarithms. First, use the product rule to rewrite the equation as \(\ln \frac{x+1}{x} = \ln 4\). Then, convert to exponential form and solve for x to get \(x = \frac{1}{3}\). This solution can be checked using a graphing calculator, by graphing the given functions and confirming that they intersect at the point \((\frac{1}{3}, \ln 4)\).

Step 1: Use the product rule for logarithms

We are given the equation: \[\ln(x + 1) - \ln x = \ln 4\] Let's use the product rule for logarithms which states that \(\ln a - \ln b = \ln \frac{a}{b}\), so: \[\ln \frac{x + 1}{x} = \ln 4\]

Step 2: Convert equation to exponential form

Since the logarithms on both sides of the equation have the same base, we can use the relationship between logarithmic and exponential forms, which states that if \(\ln a = \ln b\), then \(a = b\). Therefore, we can write our equation as: \[\frac{x + 1}{x} = 4\]

Step 3: Remove the fraction and solve for x

Now, we have an algebraic equation to solve for x. First, clear the fraction by multiplying both sides by x: \(x + 1 = 4x\) Next, subtract x from both sides: \(1 = 3x\) Then, divide both sides by 3: \(x = \frac{1}{3}\)

Step 4: Check the solution using a graphing calculator

To check the solution using a graphing calculator, graph the function \(y = \ln(x + 1) - \ln x\) and the horizontal line \(y = \ln 4\). Look for the point where the graphs intersect. If the x-coordinate of the point of intersection matches our solution, that means our answer is correct. When you graph these functions, you'll see that the point of intersection is \((\frac{1}{3}, \ln 4)\), so our solution of \(x = \frac{1}{3}\) is correct.

Key Concepts

Product Rule for Logarithms

Understanding the product rule for logarithms will help simplify complex logarithmic expressions. This rule states that the difference of two logarithmic terms with the same base can be combined into a single logarithm. Mathematically, it is expressed as:

• \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \)

In our exercise, we used this rule to combine \( \ln(x + 1) \) and \( -\ln x \). We turned them into one log term, \( \ln \left( \frac{x + 1}{x} \right) \). This is crucial because having a single logarithmic expression allows us to solve the equation more easily. Logarithms with the same base can be equated directly, making it one of the foundational skills for solving logarithmic equations.

Exponential Form

Converting logarithmic expressions to exponential form is an essential technique for solving logarithmic equations. The basic principle behind this conversion is understanding that a logarithm is an exponent. Specifically, if you have a logarithmic equation \( \ln a = \ln b \), it can be rewritten in its exponential form as:

• \( a = b \)

In our case, we transformed \( \ln \left( \frac{x + 1}{x} \right) = \ln 4 \) into the exponential equation \( \frac{x + 1}{x} = 4 \). This step is vital because it removes the logarithm, allowing us to focus on solving a simpler algebraic equation. Recognizing when to switch to exponential form helps unlock solutions that might otherwise seem complicated.

Solving Algebraic Equations

After converting our logarithmic equation into an algebraic one, the real work begins. Solving \( \frac{x + 1}{x} = 4 \) involves a few key algebraic steps. Here's how it works:

• First, eliminate the fraction by multiplying both sides by \( x \). This gives us the equation \( x + 1 = 4x \).
• Next, isolate the variable by subtracting \( x \) from both sides, yielding \( 1 = 3x \).
• Finally, divide both sides by 3 to solve for \( x \), resulting in \( x = \frac{1}{3} \).

Each of these steps is aimed at simplifying and isolating the variable to one side of the equation. Once you have isolated \( x \), you've successfully solved the algebraic equation. These steps demonstrate the seamless connection between logarithmic and algebraic equations, highlighting the intertwined nature of these mathematical concepts.