Problem 51 - College Algebra

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 \( x = 9 \), graphically verified by the intersection at \( x = 9 \).

Step 1: Simplify the Equation

Begin by simplifying the given equation: \( \log_{9}(x) - 5 = -4 \). Add 5 to both sides to isolate the logarithmic expression: \( \log_{9}(x) = 1 \).

Step 2: Convert from Logarithmic to Exponential Form

Convert the equation from logarithmic form to exponential form. The equation \( \log_{9}(x) = 1 \) can be rewritten as \( 9^1 = x \).

Step 3: Solve for x

Calculate the exponential expression: \( 9^1 = 9 \). Thus, \( x = 9 \).

Step 4: Verify the Solution by Graphing

To verify the solution graphically, plot the function \( y = \log_{9}(x) - 5 \) and \( y = -4 \) on the same set of axes. The point where these two graphs intersect will confirm the solution. In this case, the intersection point should be at \( x = 9 \).

Key Concepts

Solving Equations

In the exercise, we aim to find the value of \( x \) that satisfies the equation \( \log_{9}(x) - 5 = -4 \). To solve equations like this one, we need to isolate the variable of interest, which in this case is \( x \). Here's how we do it:

• First, we simplify the right-hand side of the equation. Add the number 5 to both sides: \( \log_{9}(x) - 5 + 5 = -4 + 5 \), resulting in \( \log_{9}(x) = 1 \).
• Once simplified, the equation becomes much easier to manage, focusing now on the logarithmic piece only.

In solving equations involving logarithms, the goal is always to simplify and isolate the variable step by step. By keeping the equation balanced, we ensure that our solution is both accurate and complete. Remember, solving these equations is about breaking it down into smaller, manageable parts so you can handle them with confidence.

Exponential Form

Converting from logarithmic to exponential form is a key step in solving logarithmic equations. This process helps us understand the relationship between the base, the result, and the exponent. In our equation, after simplifying, we have \( \log_{9}(x) = 1 \). This tells us that 9 raised to the power of 1 equals \( x \). Thus, we rewrite this as an exponential equation:

• \( \log_{9}(x) = 1 \) translates to \( 9^1 = x \).
• This conversion is handy because exponential equations are typically easier to evaluate directly.

After finding the exponential form, calculate the right side: \( 9^1 = 9 \), which tells us that \( x = 9 \). Conversions like these are key tools in effectively simplifying problems so you can find straightforward solutions. Remember, practice makes perfect!

Graphing Functions

Graphing can provide a visual verification of your solutions to equations. By graphing, you can see where different expressions intersect, which helps confirm that your calculated solution is correct.Here's how we apply graphing to this problem:

• First, consider the two functions derived from the equation: \( y = \log_{9}(x) - 5 \) and \( y = -4 \).
• Graph both functions on the same set of axes using a graphing calculator or software.
• The intersection point of the two graphs is where their values are equal, giving you a visual check that supports your algebraic solution.

In our case, the intersection occurs at \( x = 9 \), which matches the solution we found earlier. Graphing not only helps verify your answers but also enhances your understanding of how the functions behave and interact. Engaging with graphs can deepen your comprehension and provide valuable insights into mathematical relationships.