Problem 341 - Precalculus

Question

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: Simplify the Equation

Start with the given equation: \( \log_{9}(x) - 5 = -4 \). To simplify, add 5 to both sides to isolate the logarithmic function: \( \log_{9}(x) = 1 \).

Step 2: Solve for x

Recall that \( \log_{9}(x) = 1 \) means that \( x = 9^1 \). Therefore, \( x = 9 \). This is the solution to the equation.

Step 3: Verify the Solution Graphically

Graph the function \( y_1 = \log_{9}(x) - 5 \) and the horizontal line \( y_2 = -4 \). Both sides of the equation are represented by these graphs. The solution \( x = 9 \) corresponds to where these graphs intersect. At \( x = 9 \), both \( y_1 \) and \( y_2 \) equal \(-4\), confirming our solution.

Step 4: Conclusion

The solution to the equation \( \log_{9}(x) - 5 = -4 \) is \( x = 9 \). Graphically, the intersection verifies that this x-value is correct as the point (9,-4) lies on both functions.

Key Concepts

Graphing Functions

Graphing functions is an essential tool in mathematics, as it helps us visualize equations and their solutions. In this exercise, we're looking at the function represented by the equation \( y = \log_9(x) - 5 \), and we want to understand where this function intersects with the constant line \( y = -4 \).

Here are the steps for graphing these functions:

• Identify the components: For the primary function \( y_1 = \log_9(x) - 5 \), \( y_1 \) represents a vertical translation of the logarithmic function \( \log_9(x) \), shifted down by 5 units.
• Plot the graph: Start by plotting the logarithmic function \( \log_9(x) \) which has a base of 9. Then, apply the translation by subtracting 5, resulting in \( y_1 = \log_9(x) - 5 \).
• Draw the constant line: The line \( y_2 = -4 \) is straightforward since it's a horizontal line where \( y \) equals \(-4\) no matter the value of \( x \).

By graphing, you can see where these two lines intersect. This intersection represents the solution to the logarithmic equation.

Solving Equations

Solving equations, particularly logarithmic ones, involves manipulating the equation to find the value of the unknown variable. Let's break down the process used in this exercise:

• Isolate the logarithmic function: Start with the given equation \( \log_9(x) - 5 = -4 \). To simplify it, add 5 to both sides to isolate the logarithm: \( \log_9(x) = 1 \).
• Convert to exponential form: Recall that a logarithmic equation is another way to express exponentials. Thus, \( \log_9(x) = 1 \) implies that \( x = 9^1 \).
• Calculate the solution: Simplifying \( 9^1 \) results in \( x = 9 \). This is the value that solves the original equation.

Understanding the conversion between logarithmic and exponential forms is crucial since it allows us to solve for the variable smoothly.

Verification by Graphing

Verification by graphing provides a tangible way to affirm that our algebraic solution is correct. By graphing both sides of an equation, we can visually confirm their intersection at the solution point.

• Graph the solution: Plot the function \( y_1 = \log_9(x) - 5 \) and the line \( y_2 = -4 \). These graphs should intersect at the point where both expressions are equal, which should occur at the solution \( x = 9 \).
• Check the intersection: Upon graphing, you should find that the intersection point is exactly \( (9, -4) \). This affirms that when \( x = 9 \), both sides of the equation yield the same value of \(-4\).

Verification by graphing is a powerful method in mathematics as it visually demonstrates that the calculated solution is indeed correct, providing confidence in the workings of the equation.