Problem 50 - Beginning and Intermediate Algebra

Question

Solve the logarithmic equation algebraically. Then check using a graphing calculator. $$\ln x-\ln (x-4)=\ln 3$$

Step-by-Step Solution

Answer

Using logarithm properties, we simplify the given equation and get \(\ln \frac{x}{x-4} = \ln 3\). Solving for x, we find that \(x = 6\). Checking the solution algebraically, we find that \(\ln 3 = \ln 3\), which verifies our solution. Finally, by checking with a graphing calculator, we confirm that the graphs of \(y_1 = \ln(x) - \ln(x-4)\) and \(y_2 = \ln(3)\) intersect at \(x = 6\).

Step 1: Use logarithm properties to simplify

Using the logarithm property, \(\ln a - \ln b = \ln \frac{a}{b}\), we can rewrite the given equation: \[\ln x - \ln (x - 4) = \ln 3 \implies \ln \frac{x}{x - 4} = \ln 3\]

Step 2: Solve for x

Now that we have simplified the equation, we can eliminate the natural logarithm (ln) on both sides by using the property that if \(\ln a = \ln b\), then \(a = b\): \[\frac{x}{x - 4} = 3\] Now, solve for x by multiplying both sides by (x - 4) and rearranging the equation to get a quadratic equation: \[x = 3(x - 4) \implies x = 3x - 12 \implies 2x = 12\] Divide by 2 to get the value of x: \[x = 6\]

Step 3: Check the solution algebraically

To check the solution, plug the value of x back into the original equation and make sure the equation holds true: \[\ln 6 - \ln (6 - 4) = \ln 3\] \[\ln 6 - \ln 2 = \ln 3\] Using the logarithm property, \(\ln 6 - \ln 2 = \ln \frac{6}{2}\): \[\ln \frac{6}{2} = \ln 3\] Simplifying, we get: \[\ln 3 = \ln 3\] The original equation holds true with our calculated value of x, which is x = 6.

Step 4: Check the solution using a graphing calculator

To check the solution using a graphing calculator, graph the two equations: \(y_1 = \ln(x) - \ln(x-4)\) and \(y_2 = \ln (3)\). Look for the point of intersection of the two graphs. You should find that they intersect at x = 6, which confirms our algebraic solution.

Key Concepts

Properties of Logarithms

Logarithms have specific properties that make them very useful in simplifying complex equations. One such property is the difference of logarithms which states:

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

This property allows us to rewrite a subtraction of two logarithms as the logarithm of a division. This can transform what appears to be a complex equation into a simpler one. For example, in the problem \( \ln x - \ln(x-4) = \ln 3 \), this property helps you reframe it as \( \ln \frac{x}{x-4} = \ln 3 \).
This reformulation sets the stage for easier manipulation and solution of the equation. Understanding these properties is crucial for working with logarithmic equations effectively.

Graphing Calculators

Graphing calculators are powerful tools that can visualize complex equations or functions. When dealing with logarithmic equations, having a visual display can clarify points of intersection that represent solutions.
To check the solution of a logarithmic equation like \( \ln x - \ln(x-4) = \ln 3 \), you can graph two functions:

• Function one: \( y_1 = \ln(x) - \ln(x-4) \), and
• Function two: \( y_2 = \ln(3) \).

Look for where these two graphs intersect. The x-value at the intersection point is the solution to the equation. This graphical representation provides a secondary confirmation that an algebraic solution is correct.

Algebraic Solution

Solving logarithmic equations algebraically involves transforming the equation into a simpler form using logarithm properties. After simplifying, we compares the insides of the logarithms on both sides.
For the given problem, once it is simplified to \( \ln \frac{x}{x-4} = \ln 3 \), you directly equate the expressions:

• \( \frac{x}{x-4} = 3 \).

You can solve this equation by eliminating the fraction and rearranging the terms to isolate x. This leads to dividing by the coefficient of x, resulting in an effective solution. In our example, after rearranging, we reach \( x = 6 \).
This step involves basic algebraic skills, highlighting how logarithmic problems can be converted into more familiar algebraic forms for solution.

Checking Solutions Graphically

Graphically checking solutions involves plotting the simplified forms of our equations. It acts as a way to visually confirm that our algebraic manipulations are correct.
Once both functions \( y_1 = \ln(x) - \ln(x-4) \) and \( y_2 = \ln(3) \) are graphed, the point where they intersect should match the algebraic solution found.

• Verify that both equations are correctly plotted.
• Identify the x-coordinate of the intersection point.

Finding x = 6 at this intersection validates the algebraic solution. The graphical method can also help identify potential errors or extraneous results provided by the algebraic computations, offering a comprehensive understanding of the problem.