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

To solve the equation \( 2 = \log_{b} 400 \), we can use the definition of logarithms. This means that \( b^2 = 400 \). Calculating \( b \), we find: \[ b = \sqrt{400} = 20 \] Now, checking the other options quickly: - For \( b = 200 \) : \( \log_{200} 400 \) does not equal 2. - For \( b = 160,000 \) : \( b^2\) would be much larger than 400. - For \( b = 800 \) : Again, \( b^2 \) would not equal 400. Therefore, the only value matching the equation is \( b = 20 \). So the answer is \( b = 20 \).