Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-24: Find the exact solution, using common logarithms, and a two- decimal-place approximation of each solution, when appropriate. $$ \log \left(x^{2}+4\right)-\log (x+2)=2+\log (x-2) $$

Step-by-Step Solution

Answer

The exact solution is \( x = \sqrt{\frac{404}{99}} \), and the two-decimal approximation is \( x \approx 2.02 \).

Step 1: Combine Logarithms on the Left Side

Apply the logarithm property \( \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \) to combine the logarithms on the left side of the equation:\[ \log\left(\frac{x^2 + 4}{x + 2}\right) = 2 + \log(x - 2) \]

Step 2: Remove Logarithms by Exponentiating

To remove the logarithms, exponentiate both sides of the equation to convert the log equation into an exponential equation:\[ \frac{x^2 + 4}{x + 2} = 10^2 \times (x - 2) \]\[ \frac{x^2 + 4}{x + 2} = 100(x - 2) \]Ensure to handle the condition \( x > 2 \) as the logarithm \( \log(x - 2) \) is defined.

Step 3: Clear the Fraction and Simplify

Multiply both sides of the equation by \(x + 2\) to get rid of the fraction:\[ x^2 + 4 = 100(x - 2)(x + 2) \]Use the difference of squares formula to expand the right side:\[ x^2 + 4 = 100(x^2 - 4) \]

Step 4: Solve the Quadratic Equation

Rearrange the equation to form a quadratic equation:\[ x^2 + 4 = 100x^2 - 400 \]\[ 0 = 99x^2 - 404 \]Divide the entire equation by 99 to simplify:\[ x^2 = \frac{404}{99} \]Take the square root of both sides:\[ x = \pm \sqrt{\frac{404}{99}} \]

Step 5: Calculate Numerical Solution

Calculate the approximate value of \( \sqrt{\frac{404}{99}} \) using a calculator:\[ x \approx \sqrt{\frac{404}{99}} \approx 2.02 \]Remember that only the positive root is valid (since \( x > 2 \)), so discard the negative root.

Key Concepts

Logarithm Properties

Logarithms have some powerful properties that can greatly simplify complex equations. One of these properties allows us to combine or separate logarithms, making it easier to solve equations. For instance, the property \( \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \) is particularly useful when dealing with logarithmic expressions. This means if you have the log of a quotient or the difference of logs, you can convert it to a single logarithmic expression.
This consolidated expression can help in further steps to solve the equation.
In the given problem, recognizing and applying these properties correctly is crucial in simplifying and solving the equation. It helps turn a seemingly complex problem into a more manageable form.

• Combining logarithms reduces the number of logarithmic terms.
• This transformation is a standard approach to make solving easier.

Exponential Equations

Once logarithms are combined, the next step often involves converting the logarithmic equation into an exponential form. This process simplifies the equation further. Exponentiating both sides of an equation with logarithms removes the logs, changing the problem from a logarithmic to an exponential equation.
In our solution, we move from the logarithmic equation \( \log\left(\frac{x^2 + 4}{x + 2}\right) = 2 + \log(x - 2) \) to the exponential form \( \frac{x^2 + 4}{x + 2} = 100(x - 2) \).
This is done by recognizing that the base of common logarithms is 10 and raising 10 to the power of both sides of the equality.

• Exponentiating is a key step in solving logarithmic equations.
• The base of the common logarithm is 10, crucial for exponentiation.

By changing the logarithmic equation to its equivalent exponential form, we eliminate the logarithm and can proceed to solve the ensuing expression more directly.

Quadratic Equations

After transforming the equation into an exponential form, the next task is to simplify further, usually leading to a quadratic equation. Quadratic equations, which involve terms such as \( ax^2 + bx + c = 0 \), can be solved using a variety of methods, including factoring, completing the square, or using the quadratic formula. In this problem, we deal with a quadratic expression that arises naturally from the simplification process.
We arrived at the equation \( 99x^2 - 404 = 0 \), which simplifies to \( x^2 = \frac{404}{99} \).
Solving this involves taking the square root of both sides to find the values of \( x \).

• Quadratic solutions often yield two roots: positive and negative.
• Considering any predetermined conditions (like \( x > 2 \)) is essential in selecting the correct root.

This highlights the importance of both understanding quadratic equations and how to manipulate them, as they frequently appear in various mathematical contexts after such transformations.

Numerical Approximation

In mathematics, an exact solution isn't always practical or necessary, especially in real-world applications. This is where numerical approximations come into play. Once the quadratic equation is solved, it's often useful to provide a decimal approximation of the solution, which can offer a clearer understanding of the result.
For our example, the solution \( x = \pm \sqrt{\frac{404}{99}} \) was approximated to \( x \approx 2.02 \).
This was determined using a calculator to find a numerical expression of the square root.

• Numerical approximations are useful for easier interpretation of results.
• They help verify whether the solutions satisfy any applied constraints.

Thus, while exact mathematical solutions are critical, approximations provide a practical way of understanding and applying these solutions effectively.