Questions: Fill in the missing values to make the equations true. (a) log base 8 of 7 - log base 8 of 11 = log base 8 of square (b) log base 3 of 8 + log base 3 of square = log base 3 of 88 (c) log base 8 of 25 = square log base 8 of 5

Solution Steps

The equation given is: \[ \log_{8} 7 - \log_{8} 11 = \log_{8} \square \] Using the logarithm subtraction rule \(\log_{a} b - \log_{a} c = \log_{a} \left(\frac{b}{c}\right)\), we can rewrite the left-hand side: \[ \log_{8} 7 - \log_{8} 11 = \log_{8} \left(\frac{7}{11}\right) \] Thus, the missing value is \(\frac{7}{11}\).

The equation given is: \[ \log_{3} 8 + \log_{3} \square = \log_{3} 88 \] Using the logarithm addition rule \(\log_{a} b + \log_{a} c = \log_{a} (b \cdot c)\), we can rewrite the left-hand side: \[ \log_{3} 8 + \log_{3} \square = \log_{3} (8 \cdot \square) \] Set this equal to \(\log_{3} 88\): \[ \log_{3} (8 \cdot \square) = \log_{3} 88 \] Since the logarithms are equal, their arguments must be equal: \[ 8 \cdot \square = 88 \] Solving for \(\square\): \[ \square = \frac{88}{8} = 11 \]

The equation given is: \[ \log_{8} 25 = \square \cdot \log_{8} 5 \] We know that \(25 = 5^2\), so: \[ \log_{8} 25 = \log_{8} (5^2) \] Using the logarithm power rule \(\log_{a} (b^c) = c \cdot \log_{a} b\), we can rewrite the left-hand side: \[ \log_{8} (5^2) = 2 \cdot \log_{8} 5 \] Thus, the missing value is \(2\).

Final Answer