Problem 51 - 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. \(\log _{9}(x)-5=-4\)

Step-by-Step Solution

Answer

The solution is \( x = 9 \).

Step 1: Isolate the logarithmic term

The equation given is \( \log_{9}(x) - 5 = -4 \). To isolate the logarithmic term, add 5 to both sides of the equation. This gives you \( \log_{9}(x) = 1 \).

Step 2: Solve the logarithm

We have \( \log_{9}(x) = 1 \). By the definition of logarithms, this means that \( 9^{1} = x \). Therefore, \( x = 9 \).

Step 3: Verification through graphing

Graph \( y = \log_{9}(x) - 5 \) and \( y = -4 \) on the same set of axes. The point of intersection of these graphs, if it exists, should occur at \( x = 9 \).

Step 4: Checking the intersection

The graph of \( y = \log_{9}(x) - 5 \) should intersect the horizontal line \( y = -4 \) at the point \( (9, -4) \). This confirms our solution. Thus, the solution to the equation is verified as correct.

Key Concepts

Logarithmic Functions

Logarithmic functions are essential in solving equations where a variable appears within a logarithm. The fundamental understanding you need is that a logarithm is the inverse operation of exponentiation. For instance, the equation \( \log_{b}(x) = y \) means that \( b^{y} = x \). Knowing this relationship allows you to "undo" a logarithm by exponentiating. When you encounter an equation like \( \log_{9}(x) = 1 \), you can rewrite this as \( 9^1 = x \), resulting in \( x = 9 \). This step is crucial when you are tasked to isolate and solve for the variable inside the logarithm.
This is the basic process when working with logarithmic equations:

• Identify and isolate the logarithmic term.
• Apply the fundamental definition of logarithms to solve for the unknown variable.

By mastering this simple process, solving logarithmic equations becomes a straightforward task.

Graphing Equations

Graphing is a powerful tool for visualizing solutions to equations such as the one presented in our example. When graphing, you represent each side of the equation as a function on a coordinate plane. In this example, you graph \( y = \log_{9}(x) - 5 \) and \( y = -4 \).
By graphing, you look for the point where these functions intersect, providing a visual confirmation of the solution. This is significant because:

• The point of intersection corresponds to the values of \( x \) and \( y \) that satisfy both equations simultaneously.
• In our example, finding that point confirms the algebraic solution was executed correctly.

Graphing also provides an intuitive understanding of how logarithmic functions behave and change, pushing beyond mere number-crunching and into deeper comprehension. You can see how changes in the logarithmic base or constant terms translate geometrically.

Verification of Solutions

Verifying solutions is a critical step that ensures mathematical robustness. After solving the logarithmic equation algebraically, the next step is to check that solution graphically. When the graphs intersect at the solved value (like \( x = 9 \) in our example), it validates your work.
Here are a few tips for effective verification:

• Always plot accurately using appropriate software or a graphing calculator.
• Check that the intersection point's coordinates align with your algebraic findings.

This process doesn't just confirm your answer, but it also aligns your algebraic manipulations with a tangible graphical proof. The combination of algebraic and graphical solutions enhances comprehension and ensures the solution's reliability and validity.