Questions: Solve the equation. log base x of 16 equals 2 Select one: a. 2 b. 4,-4 c. 2,-2 d. 4 Solve the equation. log base 3 of 1/27 equals x Select one: a. 3 b. -3 c. 1/81 d. 1/9

Solution Steps

To solve these logarithmic equations, we need to convert the logarithmic form to its exponential form and then solve for the variable.

1. For the equation \(\log_{x} 16 = 2\):

   • Convert the logarithmic equation to its exponential form: \(x^2 = 16\).
   • Solve for \(x\).

2. For the equation \(\log_{3} \frac{1}{27} = x\):

   • Convert the logarithmic equation to its exponential form: \(3^x = \frac{1}{27}\).
   • Solve for \(x\).

To solve the equation \( \log_{x} 16 = 2 \), we convert it to its exponential form: \[ x^2 = 16 \] Taking the square root of both sides, we find: \[ x = \sqrt{16} \quad \text{or} \quad x = -\sqrt{16} \] Thus, the solutions are: \[ x = 4 \quad \text{and} \quad x = -4 \]

Next, we solve the equation \( \log_{3} \frac{1}{27} = x \) by converting it to exponential form: \[ 3^x = \frac{1}{27} \] Recognizing that \( 27 = 3^3 \), we can rewrite the equation as: \[ 3^x = 3^{-3} \] This implies: \[ x = -3 \]

Final Answer