Problem 61 - College Algebra with Corequisite Support

Question

For the following exercises, solve the equation for \(x\), if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log _{9}(3-x)=\log _{9}(4 x-8)\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{11}{5}\).

Step 1: Equate the Arguments of the Logarithms

Since the bases of the logarithms on both sides of the equation are the same, we can equate the arguments. Thus, we have the equation: \[3 - x = 4x - 8\]

Step 2: Rearrange the Equation

Move all the terms involving \(x\) to one side of the equation. Start by adding \(x\) to both sides:\[3 = 5x - 8\]Next, add 8 to both sides to isolate terms with \(x\):\[11 = 5x\]

Step 3: Solve for x

Divide both sides by 5 to solve for \(x\):\[x = \frac{11}{5}\]

Step 4: Verify the Domain of Logarithm

Logarithms are defined only for positive arguments. Thus, the arguments of the logarithms, \(3-x\) and \(4x-8\), must be positive.For \(3-x > 0\): \(x < 3\)For \(4x-8 > 0\): \(x > 2\)Thus, the valid domain for \(x\) is \(2 < x < 3\). Since \(x = \frac{11}{5} = 2.2\), it lies within this range.

Step 5: Graph Both Sides

Plot \(y = \log_9(3-x)\) and \(y = \log_9(4x-8)\) on the same set of axes. The point of intersection of these two graphs will verify the solution. Observe that they intersect at \(x = \frac{11}{5}\), confirming the solution.

Key Concepts

Graphing Functions

Graphing functions helps us visually understand equations and their solutions. For the given problem, we graph the functions on either side of the equation: \(y = \log_{9}(3-x)\) and \(y = \log_{9}(4x-8)\). Graphing these functions on a coordinate plane allows us to find the point at which the equations are equal. This point of intersection corresponds to the solution found algebraically.

To graph these functions, complete the following steps:

• Create a table of values by selecting points within the domain of each logarithmic function.
• Plot each value on the graph to create their respective curves.
• Identify where the graphs intersect. This point confirms the algebraic solution.

When plotting, ensure both functions lie within their respective domains to accurately depict their behavior. The intersection confirms that our solution \(x = \frac{11}{5}\) is correct.

Domain of a Function

The domain of a function is critical in solving equations, particularly when dealing with logarithms. The domain refers to all possible input values (\(x\)) that do not invalidate the function.

Logarithmic functions, like those in the given problem, are defined only for positive arguments. For \(\log_{9}(3-x)\), the inequality\(3-x > 0\) restricts \(x\) to values less than 3, so \(x < 3\). On the other hand, for \(\log_{9}(4x-8)\), \(4x-8 > 0\) means \(x > 2\).

Combining these inequalities gives a domain intersection: \(2 < x < 3\), which specifies the valid range of \(x\) values for solutions. Proper identification of domains ensures our potential solutions do not create undefined situations in our equations. Thus, verifying solutions within the valid domain is crucial for correctness.

Solving Equations

Solving equations involves finding the values of \(x\) that satisfy the given conditions. In our logarithmic equation, \(\log_{9}(3-x) = \log_{9}(4x-8)\), both logarithms share the same base. Hence, their arguments can be set equal for solutions, leading to \(3 - x = 4x - 8\).

Solving requires basic algebraic manipulations:

• First, move all \(x\)-related terms to one side: add \(x\) to both sides resulting in \(3 = 5x - 8\).
• Then, isolate the 5\(x\) term by adding 8 to both sides, giving \(11 = 5x\).
• Finally, divide by 5 across to solve for \(x\), resulting in \(x = \frac{11}{5}\).

Check this solution against the domain restrictions (\(2 < x < 3\)). Since \(\frac{11}{5} = 2.2\), it fits within bounds. Verifying both algebraically and graphically provides confirmation that \(x = \frac{11}{5}\) is the correct solution.