Sure, let's carefully solve each part of the question step-by-step:
### Part (b)
Given:
[tex]\[ \log_5 11 - \log_5 \square = \log_5 \frac{11}{4} \][/tex]
We can use the properties of logarithms to solve this equation. One key property to remember is:
[tex]\[ \log_b a - \log_b c = \log_b \left( \frac{a}{c} \right) \][/tex]
Using this property, we can rewrite the left-hand side of the given equation:
[tex]\[ \log_5 11 - \log_5 \square = \log_5 \left( \frac{11}{\square} \right) \][/tex]
According to the given equation:
[tex]\[ \log_5 \left( \frac{11}{\square} \right) = \log_5 \frac{11}{4} \][/tex]
Since the logarithms are equal, their arguments must be equal:
[tex]\[ \frac{11}{\square} = \frac{11}{4} \][/tex]
By equating these fractions:
[tex]\[ \square = 4 \][/tex]
So, the solution for part (b) is:
[tex]\[ \square = 4 \][/tex]
### Part (c)
Given:
[tex]\[ 4 \log_8 2 = \log_8 \square \][/tex]
We can use another property of logarithms here:
[tex]\[ n \log_b a = \log_b (a^n) \][/tex]
Applying this property to the left-hand side of the given equation:
[tex]\[ 4 \log_8 2 = \log_8 (2^4) \][/tex]
Hence, we can rewrite the equation as:
[tex]\[ \log_8 (2^4) = \log_8 \square \][/tex]
Since the logarithms are equal, their arguments must be equal too:
[tex]\[ 2^4 = \square \][/tex]
Calculating the value on the right-hand side:
[tex]\[ 2^4 = 16 \][/tex]
Thus, the solution for part (c) is:
[tex]\[ \square = 16 \][/tex]
In summary, the solutions are:
- For part (b): [tex]\( \square = 4 \)[/tex]
- For part (c): [tex]\( \square = 16 \)[/tex]
