Problem 130 - College Algebra Essentials

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. $$\log (x-15)+\log x=2$$

Step-by-Step Solution

Answer

The solution to this equation is the \(x\) values where the graph of the function \(x^2 - 16x + 100 = 0\) intersects with \(x\)-axis. This \(x\) value is verified via substitution into the original equation \(log (x-15) + log x = 2\).

Step 1: Convert the Logarithmic Equation

First of all, it's necessary to convert the current logarithmic equation in a form that can be easily graphed. Apply the property of log: \(log A + log B = log(AB)\) to the given equation. This will turn into: \(log((16-x)(x)) = 2\).

Step 2: Convert to Exponential Form

The next step is to convert the new equation in the exponential form. The equivalent expression in exponential form (for \(log_ba = c\) is \(b^c=a\)) is \((16-x)(x) = 10^2\), which simplifies to \((16-x)(x) = 100\). Now, this function is easier to take to the graphing utility.

Step 3: Rearrange and graph

Rearrange equation from step 2 to: \((x)(x-16)= -100\). It opens up to \(x^2 - 16x + 100 = 0\). Now, graph this function using a graphing utility. The intersection with \(x\) -axis are solutions to equation.

Step 4: Determine the x-coordinate of the Intersection

Look for the \(x\)-coordinates of the intersection points. These points represent the solutions to the equation. Use the regression feature of your graphing utility to get accurate \(x\) values.

Step 5: Verify by Substituting the Values

Finally, make sure to verify these \(x\) values by substituting them in the original equation. Verify by directly substituting into: \(log (x-15) + log x = 2\). Both sides should result into a same value.

Key Concepts

Logarithmic Properties

Logarithmic properties are fundamental when working with equations that involve logarithms. They help simplify and transform equations into more manageable forms. One key property is the product rule, which states that \(\log_b(A) + \log_b(B) = \log_b(AB)\). This rule allows you to combine logarithms with the same base into a single logarithm term.
This property is particularly useful for solving equations like \(\log(x-15) + \log x = 2\). Here, by applying the product rule, you can simplify the equation to \(\log((x-15)x) = 2\). Using this simplified form makes it easier to perform further operations on the equation. Keep in mind that other logarithmic properties, like the quotient rule and power rule, can further assist in manipulating and solving logarithmic expressions.

Exponential Form Conversion

Converting logarithmic equations into exponential form is an essential step in solving them, particularly when you need to graph them. The conversion is based on the fact that if you have \(\log_b(A) = c\), then you can express the same relationship as \(b^c = A\).
In our example, after simplifying the equation with the logarithmic properties, we have \(\log((x-15)x) = 2\). Converting this to an exponential form, knowing that the base of the logarithm is 10, results in \((x-15)x = 10^2\), or \((x-15)x = 100\). This exponential form allows for a straightforward graphing process, as it becomes a standard polynomial equation that many graphing utilities can easily plot.

Graphing Utilities

Graphing utilities, such as graphing calculators or software, are powerful tools in visualizing equations and finding their solutions. They allow you to input complex expressions and display the graphs, providing a clear picture of how the equation behaves across different values of \(x\).
For the equation \(x(x-16) = -100\), after converting it from its logarithmic form, using a graphing utility makes it easier to identify the intersection points with the \(x\)-axis. This intersection gives the solutions to the equation. Graphing utilities often include features like zoom and range settings, which allow for precise adjustments to see the graph accurately and identify solutions more easily.

Verifying Solutions

Verifying solutions is a crucial step to ensure the correctness of the answer obtained from graphing or algebraic manipulation. Once you find potential solutions by identifying the intersection points on the graph, you are not yet finished. The final step is to substitute these solutions back into the original equation to check if they satisfy it.
For the equation \(\log(x-15) + \log x = 2\), substitute the \(x\)-values obtained from the graph back into this equation. Calculate each term separately and ensure both sides of the equation equal each other. If they do, your solution is verified. If not, you may need to check your graph or calculations for errors. This process confirms that the solutions are accurate and applicable to the problem.