Solve: \( 2=\log _{b} 400 \) \( b=20 \) \( b=200 \) \( b=160,000 \) \( b=800 \)

To solve the equation: \[ 2 = \log_b 400 \] **Step 1:** Understand the logarithmic equation. The equation \( \log_b 400 = 2 \) can be rewritten in its exponential form using the definition of a logarithm: \[ b^2 = 400 \] **Step 2:** Solve for \( b \). To find the value of \( b \), take the square root of both sides: \[ b = \sqrt{400} \] \[ b = 20 \] **Important Note:** In logarithms, the base \( b \) must be a positive real number and not equal to 1. Therefore, we discard the negative root (i.e., \( b = -20 \)) because it does not satisfy the conditions for a logarithmic base. **Conclusion:** The correct value of \( b \) is: \[ b = 20 \] **Given Options:** - \( b = 20 \) - \( b = 200 \) - \( b = 160{,}000 \) - \( b = 800 \) **Correct Answer:** \( b = 20 \)