Questions: Fill in the missing values to make the equations true. (a) log2 11 + log2 5 = log2 □ (b) log3 □ - log3 4 = log3 7/4 (c) log9 32 = □ log9 2

Solution Steps

The logarithm addition rule states that \( \log_b a + \log_b c = \log_b (a \cdot c) \).
Apply this rule to part (a):
\[ \log_2 11 + \log_2 5 = \log_2 (11 \cdot 5) = \log_2 55. \] Thus, the missing value is \( 55 \).

The logarithm subtraction rule states that \( \log_b a - \log_b c = \log_b \left( \frac{a}{c} \right) \).
Apply this rule to part (b):
\[ \log_3 x - \log_3 4 = \log_3 \left( \frac{x}{4} \right). \] Set this equal to \( \log_3 \frac{7}{4} \):
\[ \log_3 \left( \frac{x}{4} \right) = \log_3 \frac{7}{4}. \] Since the logarithms are equal, their arguments must be equal:
\[ \frac{x}{4} = \frac{7}{4}. \] Solve for \( x \):
\[ x = 7. \] Thus, the missing value is \( 7 \).

The logarithm power rule states that \( \log_b a^k = k \log_b a \).
Rewrite \( \log_9 32 \) in terms of \( \log_9 2 \):
\[ \log_9 32 = \log_9 (2^5) = 5 \log_9 2. \] Thus, the missing value is \( 5 \).

Final Answer