Problem 127 - Algebra and Trigonometry

Question

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$ \log _{3}(4 x-7)=2 $$

Step-by-Step Solution

Answer

The solution set to the equation \( \log _{3}(4 x-7)=2 \) is \(x = 4\), verified both graphically and by direct substitution.

Step 1: Rearrange the Equation

First write the equation in exponential form for easier handling. This is done by letting \(3^{2} = 4x - 7\), which simplifies to \(9 = 4x - 7\).

Step 2: Solve for \(x\)

Solve for \(x\) from the equation \(9 = 4x - 7\). Do this by adding \(7\) to both sides of the equation, getting \(16 = 4x\). Divide both sides by \(4\) to obtain \(x = 4\).

Step 3: Graph the Equations

Now graph both sides of the original equation, which are \(y = \log _{3}(4 x-7)\) and \(y = 2\) on the same graph.

Step 4: Find the Intersection Point

The intersection of the two graphs represents the solution to the equation. Given the graphics, it is observed that they intersect at \(x = 4\).

Step 5: Verify the Solution

The final step is to verify the answer by substituting \(x = 4\) into the original equation. This yields \( \log _{3}(4 * 4 -7) = \log _{3}(9) = 2 \), which matches the right hand side of the equation. So, \(x = 4\) is indeed the solution to the equation.

Key Concepts

Graphing Utilities in Algebra

Graphing utilities are an invaluable tool in algebra as they help visualize equations and their solutions. These tools allow us to plot the graphs of equations and see where they intersect, which can often represent the solution to the equation.
In the context of solving logarithmic equations, using a graphing utility involves plotting the graph of each side of the equation separately. For example, in the equation \( \log_{3}(4x-7)=2 \), you would plot \( y = \log_{3}(4x-7) \) and \( y = 2 \). The intersection of these two graphs is vital, as the \( x \)-coordinate of this intersection provides the solution to the equation.
This visual method can be extremely helpful as it not only indicates the solution but also offers a deeper understanding of how the relationship between variables is expressed graphically. You can often discover approximate solutions or confirm algebraic solutions through this visual approach.

• Ensure the correct viewing rectangle to fit both graphs comfortably.
• Use the intersection point(s) to identify potential solutions.

Exponential Form Conversion

Transforming an equation from logarithmic to exponential form is a crucial step in simplifying the problem. This conversion makes it easier to solve for variables such as \( x \).
In the equation \( \log_{3}(4x-7)=2 \), converting to exponential form involves writing it as \( 3^{2} = 4x - 7 \). Here, we use the property of logarithms that states if \( \log_{b}(A) = C \), then \( b^{C} = A \).
Once in exponential form, the equation becomes simpler to work with, as basic algebraic techniques can be applied to isolate the variable \( x \). In this example, simplifying \( 3^2 \) gives \( 9 \), leading to the equation \( 9 = 4x - 7 \). Solving this step-by-step leads us closer to the solution.
Remember:

• Exponential forms can often simplify complex logarithmic expressions.
• This process can turn an abstract logarithmic problem into a manageable algebraic form.

Verifying Solutions

Verification is the final and an important step that confirms the accuracy of your solution. It involves plugging the obtained value back into the original equation and checking for consistency.
For the problem \( \log_{3}(4x-7)=2 \), once we derived \( x = 4 \), verification is done by substituting \( x \) back into the original equation. This means calculating \( \log_{3}(4 \times 4 -7) \), which simplifies to \( \log_{3}(9) \). This evaluates to 2, confirming the right side of the equation. Therefore, \( x = 4 \) is indeed a correct solution.
Verification ensures that no errors were made during the computation process. It's always good practice in math to validate your results this way.
Key tips:

• Ensure substitution is accurate and check every step of the simplification.
• If the original equation is satisfied, the solution is verified and correct.