Questions: Solve the logarithmic equation. 2 log3(4-x) - log3 5 = log3 45 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. x= (Simplify your answer. Use a comma to separate answers as needed.) B. There is no solution.

Solution Steps

To solve the given logarithmic equation, we can use the properties of logarithms to combine and simplify the terms. First, apply the power rule to the first term, then use the subtraction rule to combine the logarithms on the left side. Finally, equate the simplified expression to the right side and solve for \( x \).

We start with the logarithmic equation: \[ 2 \log_{3}(4 - x) - \log_{3}(5) = \log_{3}(45) \]

Using the properties of logarithms, we can rewrite the left side: \[ \log_{3}((4 - x)^2) - \log_{3}(5) = \log_{3}(45) \] This simplifies to: \[ \log_{3}\left(\frac{(4 - x)^2}{5}\right) = \log_{3}(45) \]

Since the logarithms are equal, we can set the arguments equal to each other: \[ \frac{(4 - x)^2}{5} = 45 \]

Multiplying both sides by 5 gives: \[ (4 - x)^2 = 225 \] Taking the square root of both sides results in: \[ 4 - x = 15 \quad \text{or} \quad 4 - x = -15 \] Solving these equations:

1. \( 4 - x = 15 \) leads to \( x = -11 \)
2. \( 4 - x = -15 \) leads to \( x = 19 \)

We need to ensure that the solutions do not make the logarithm undefined. For \( x = -11 \): \[ 4 - (-11) = 15 > 0 \quad \text{(valid)} \] For \( x = 19 \): \[ 4 - 19 = -15 < 0 \quad \text{(invalid)} \] Thus, the only valid solution is \( x = -11 \).

Final Answer