Problem 126 - Introductory and Intermediate Algebra for College Students 4th

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. $$\log _{3}(3 x-2)=2$$

Step-by-Step Solution

Answer

The solution to the given logarithmic equation is \(x = 11/3\), which is confirmed by both graphical representation and substitution into the original equation.

Step 1: Understand and Express the Logarithmic Equation

First, let's express the given equation \(\log _{3}(3 x-2)=2\) as \(3 x - 2 = 3^2\). This is done because logarithmic and exponential functions are inverses of each other.

Step 2: Solve for the Unknown

Now, resolve the equation obtained above for \(x\). We first simplify \(3x - 2 = 9\) to get \(3x = 11\). Dividing both sides by \(3\) gives us \(x = 11/3\).

Step 3: Graph the Function

At this point we would graph the two sides of the equation. One would be a logarithmic equation denoted by \(y=\log _{3}(3x-2)\) and the other a constant line represented by \(y=2\). The intersection point between both lines will confirm the solution obtained above.

Step 4: Verification by Substitution

Finally, we substitute the obtained solution \(x = 11/3\) into the given equation: \(\log _{3}(3(11/3)-2)\) which simplifies to \(\log_{3}(11-2) = \log_{3}(9) = 2\), in alignment with the given equation.