Problem 127 - 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. $$\log (x+3)+\log x=1$$

Step-by-Step Solution

Answer

After carrying out the steps above, the solutions obtained for the equation \(\log (x+3)+\log x=1\) will be found. As this process involves using a graphing utility, the exact solution may vary slightly depending on the utility.

Step 1: Simplify the Equation

Rewrite the logarithmic equation \(\log (x+3)+\log x=1\) in the equivalent exponential form. We can use the properties of logarithms and write \(\log (x+3)+\log x\) as \(\log ((x+3)x)\). Therefore, our equation becomes \(\log ((x+3)x)=1\). We can rewrite this in exponential form, where base is 10, as \((x+3)x = 10\), which simplifies to \(x^2 + 3x = 10\). Finally, rewrite the equation as \(x^2 + 3x - 10 = 0\).

Step 2: Use a Graphing Utility

Use a graphing utility to display graphs for the equations \(y = x^2 + 3x - 10\) and \(y = 0\). The x-coordinates of the intersection points of the two graphs are the solutions to the original equation.

Step 3: Find the Intersection Point

Notice where the two graphs intersect. There should be two points of intersection, corresponding to the solutions of the quadratic equation.

Step 4: Verification by Direct Substitution

After getting the solutions, substitute these values back into our original equation \(\log(x+3) + \log(x) = 1\) to check if both sides become equal. If they match, then the solutions found from the graph are correct.