Questions: Solve the logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. 2 log6(x+9) = log6 36 + 2 Solve the equation. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is □ B. (Simplify your answer. Use a comma to separate answers as needed.) B. There are infinitely many solutions. C. There is no solution.

Solution Steps

To solve the given logarithmic equation, we will first use the properties of logarithms to simplify and solve for \( x \). We will then check if the solution is within the domain of the original logarithmic expressions.

1. Use the property of logarithms that allows you to move the coefficient in front of the log as an exponent inside the log: \( 2 \log_b(a) = \log_b(a^2) \).
2. Simplify both sides of the equation using this property.
3. Equate the arguments of the logarithms since the bases are the same.
4. Solve the resulting equation for \( x \).
5. Verify that the solution is within the domain of the original logarithmic expressions, which requires that the argument of the logarithm is positive.

We start with the logarithmic equation: \[ 2 \log_{6}(x + 9) = \log_{6}(36) + 2 \]

Using the property \( a \log_b(c) = \log_b(c^a) \), we can rewrite the left side: \[ \log_{6}((x + 9)^2) = \log_{6}(36) + 2 \] Next, we express \( 2 \) as \( \log_{6}(6^2) = \log_{6}(36) \): \[ \log_{6}((x + 9)^2) = \log_{6}(36) + \log_{6}(36) \] This simplifies to: \[ \log_{6}((x + 9)^2) = \log_{6}(36 \cdot 36) = \log_{6}(1296) \]

Since the logarithms are equal, we can set the arguments equal to each other: \[ (x + 9)^2 = 1296 \]

Taking the square root of both sides gives: \[ x + 9 = \pm 36 \] This results in two equations:

1. \( x + 9 = 36 \) leading to \( x = 27 \)
2. \( x + 9 = -36 \) leading to \( x = -45 \)

The original logarithmic expression requires that \( x + 9 > 0 \):

• For \( x = 27 \): \( 27 + 9 = 36 > 0 \) (valid)
• For \( x = -45 \): \( -45 + 9 = -36 < 0 \) (invalid)

Thus, the only valid solution is \( x = 27 \).

Final Answer