Problem 63 - 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. $$ \frac{3}{\log _{2}(10)}-\log (x-9)=\log (44) $$

Step-by-Step Solution

Answer

The solution is \( x \approx 9.1818 \).

Step 1: Express log functions in terms of base 10 or change of base

Start by expressing all logarithmic terms using the same base. The term \( \log_{2}(10) \) can be expressed in terms of base 10 using the change of base formula: \( \log_{2}(10) = \frac{\log_{10}(10)}{\log_{10}(2)} = \frac{1}{\log_{10}(2)} \). The equation becomes \( \frac{3 \cdot \log_{10}(2)}{1} - \log_{10}(x-9) = \log_{10}(44) \).

Step 2: Simplify the equation

Now substitute \( \frac{3}{\log_{2}(10)} \) with \( 3 \cdot \log_{10}(2) \) and rewrite the equation: \[3 \cdot \log_{10}(2) - \log_{10}(x-9) = \log_{10}(44)\].

Step 3: Combine logarithmic terms

Using properties of logarithms, such as \( a \log(b) = \log(b^a) \), simplify \( 3 \cdot \log_{10}(2) = \log_{10}(2^3) = \log_{10}(8) \). Now the equation is \( \log_{10}(8) - \log_{10}(x-9) = \log_{10}(44) \).

Step 4: Use logarithmic subtraction rule

Apply the rule \( \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \) to simplify the left side: \( \log_{10}\left(\frac{8}{x-9}\right) = \log_{10}(44) \).

Step 5: Equalize the arguments

Since the logarithms with the same base must have equal arguments, set \( \frac{8}{x-9} = 44 \). Solve this equation by cross-multiplying: \( 8 = 44(x-9) \).

Step 6: Solve for x

Simplify \( 8 = 44x - 396 \). Add 396 to both sides to get \( 404 = 44x \). Then divide both sides by 44 to find \( x = \frac{404}{44} = 9.1818 \).

Step 7: Graph and Check Solution

To verify the solution, graph \( y = \frac{3}{\log_{2}(10)} - \log_{10}(x-9) \) and \( y = \log_{10}(44) \) on the same set of axes. Check that they intersect at \( x \approx 9.1818 \), confirming the solution.

Key Concepts

Change of Base Formula

The change of base formula is a powerful technique when dealing with logarithms. It allows you to convert a logarithm from one base to another, which is particularly helpful for solving equations or when your calculator only supports logarithms in a specific base like base 10 or base e.
For instance, if you have a log base 2 such as \( \log_{2}(10) \), you can rewrite it using the change of base formula:

• \( \log_{2}(10) = \frac{\log_{10}(10)}{\log_{10}(2)} \)
• Using base 10 log, this simplifies to \( \frac{1}{\log_{10}(2)} \)

This conversion is useful because it makes calculations more straightforward, especially when integrating them into equations that predominantly use a different base. In our given problem, it allowed us to align the logarithms under a single base for easy solving.

Properties of Logarithms

Understanding the properties of logarithms is crucial when manipulating and simplifying equations. Several key properties are frequently used:

• **Product Rule:** \( \log(a \cdot b) = \log(a) + \log(b) \)
• **Quotient Rule:** \( \log(\frac{a}{b}) = \log(a) - \log(b) \)
• **Power Rule:** \( a \cdot \log(b) = \log(b^a) \)

In our example equation, these rules helped transform and simplify expressions. For example, we used the power rule to convert \( 3 \cdot \log_{10}(2) \) to \( \log_{10}(2^3) = \log_{10}(8) \). We further used the quotient rule to simplify \( \log_{10}(8) - \log_{10}(x-9) = \log_{10}(44) \) by setting up the equation \( \log_{10}(\frac{8}{x-9}) = \log_{10}(44) \). These properties allow us to combine or break apart logarithmic terms efficiently.

Graphing Functions

Graphing logarithmic functions is a step that helps visually confirm solutions. By plotting each side of the equation on a graph, you get a visual representation of where they intersect, which represents the solution.
In our case, after finding that \( x \approx 9.1818 \), it is vital to plot:

• \( y = \frac{3}{\log_{2}(10)} - \log_{10}(x-9) \)
• \( y = \log_{10}(44) \)

These graphs should intersect at the x-value of \( 9.1818 \). This intersection confirms that our algebraic solution is correct. Graphing provides not only verification but also insight into the behavior of the logarithmic equation and how each component impacts the solution.

Verifying Solutions

Verification is a critical step in solving equations to ensure the solution found is accurate and reasonable. After solving for \( x \approx 9.1818 \) using algebra, it is important to confirm this result visually and analytically.
First, substitute \( x = 9.1818 \) back into the original equation

• Check if both sides equal when \( x = 9.1818 \).
• If both sides equal, this confirms that \( x \) is a correct solution.

Second, by graphing both sides of the equation, as already explained, ensure these graphs intersect at \( x \approx 9.1818 \).
Verifying solutions not only confirms accuracy but also develops a deeper understanding of how the solution fits within the context of the problem. This approach provides certainty and instills confidence in the problem-solving process.