Problem 73 - College Algebra

Question

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 2 \log _{3}(x+4)=\log _{3} 9+2 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 5\).

Step 1: Simplify the equation

Firstly, the equation can be simplified by converting the numerical logarithm expression on the right side to its simplest form. The \(\log_{3}9\) can be simplified to 2, as \(3^2\) equals to 9.

Step 2: Simplify further

Substitute the simplified \(\log_{3}9\) into the equation: \(2 \log_{3}(x+4) = 2 + 2\). Combine like terms on the right side of the equation to get: \(2 \log_{3}(x+4) = 4\)

Step 3: Isolate the logarithmic expression

To isolate the logarithmic expression, divide both sides of the equation by 2. This leads to: \(\log_{3}(x+4) = 2\)

Step 4: Convert the logarithmic equation to an exponential equation

The equation can be rewritten in exponential form: \(3^2 = x + 4\) or \(9 = x + 4\)

Step 5: Solve for x

Finally, solve for \(x\) by subtracting 4 from both sides of the equation: \(x = 9 - 4 = 5\)

Step 6: Confirm the validity of the solution

Substitute \(x = 5\) into the original equation to ensure it is valid: \(2 \log_{3}(5+4) = \log_{3} 9 + 2\). Simplify the equation on both sides to verify that it holds true.

Key Concepts

Exponential Form

Understanding how to convert logarithmic equations to exponential form helps solve these types of equations. A logarithmic equation like \( \log_{b}(y) = x \) can be written as \( b^x = y \). This transformation is crucial because it allows us to easily solve for the unknown variable. In our exercise, we converted the equation \( \log_{3}(x+4) = 2 \) into its exponential form \( 3^2 = x + 4 \). This makes finding the value of \( x \) straightforward, leading to the solution \( x = 5 \). Converting logarithms to exponential form simplifies problem-solving and aids in understanding the relationship between the components involved.

Domain of Logarithmic Expressions

Logarithmic functions have specific restrictions, known as domains, which must be considered when solving equations. The domain of a logarithmic function \( \log_{b}(x) \) is \( x > 0 \). This means the argument of the logarithm, \( x+4 \) in this case, must be greater than zero. By setting \( x+4 > 0 \), we find that \( x > -4 \).

• These domain rules ensure the logarithm is defined for real numbers.
• Ignoring these restrictions can lead to invalid solutions.

In the exercise, checking the domain ensured that our solution \( x = 5 \) was valid, as it satisfies \( x > -4 \). Always consider the domain to confirm the solution's legitimacy.

Exact and Decimal Approximations

Finding solutions in both exact and decimal forms is common in mathematics. The exact form is usually in integers or fractions, while the decimal approximation uses numbers rounded to a certain decimal place. In this problem, after converting the logarithmic equation, we found the exact solution \( x = 5 \). However, some exercises might require using a calculator for a decimal approximation.

• Decimal approximations are often necessary for answers involving irrational numbers or when an exact form isn't sufficient.
• In this task, the exact and decimal forms were the same, but using a calculator for verification ensures clarity.

This distinction is essential in exams or practical situations where specific formats are requested.

Valid Solution Verification

Once a solution is obtained, it's vital to verify it within the context of the original problem. This involves substituting the solution back into the original equation to check for consistency. In this exercise, substituting \( x = 5 \) back into the original equation \( 2 \log_{3}(5+4) \) confirmed the solution was correct.

• Verification avoids mistakes like calculation errors or misinterpretations of the question.
• This practice reinforces accuracy and reliability in solutions.

Rechecking the result ensures it aligns with the domain and the transformed equations throughout the solving process, bringing peace of mind and correctness to mathematical problem-solving.