To evaluate the expression [tex]\( 3 \log_2 4 + \frac{1}{2} \log_2 6 - \frac{1}{2} \log_2 24 \)[/tex], we will break it down step by step and simplify it.
### Step 1: Evaluate each logarithmic term.
Term 1: [tex]\( 3 \log_2 4 \)[/tex]
Using the property of logarithms that [tex]\( \log_b (a^c) = c \log_b a \)[/tex], we can simplify:
[tex]\[ \log_2 4 = \log_2 (2^2) = 2 \][/tex]
So,
[tex]\[ 3 \log_2 4 = 3 \times 2 = 6 \][/tex]
Term 2: [tex]\( \frac{1}{2} \log_2 6 \)[/tex]
[tex]\[
\frac{1}{2} \log_2 6
\][/tex]
This term is evaluated as a decimal:
[tex]\[ \frac{1}{2} \log_2 6 \approx 1.292481250360578 \][/tex]
Term 3: [tex]\( \frac{1}{2} \log_2 24 \)[/tex]
[tex]\[
\log_2 24 = \log_2 (2^3 \cdot 3) = 3 \log_2 2 + \log_2 3 = 3 + \log_2 3
\][/tex]
So,
[tex]\[ \frac{1}{2} \log_2 24 = \frac{1}{2} (3 \log_2 2 + \log_2 3) = \frac{1}{2} (3 + \log_2 3) \][/tex]
Approximating this,
[tex]\[ \frac{1}{2} \log_2 24 \approx 2.292481250360578 \][/tex]
### Step 2: Combine all the evaluated terms.
Now sum up the values of the terms from the previous steps:
[tex]\[
6 + 1.292481250360578 - 2.292481250360578
\][/tex]
### Step 3: Simplify the result.
Perform the arithmetic operations:
[tex]\[ 6 + 1.292481250360578 = 7.292481250360578 \][/tex]
[tex]\[ 7.292481250360578 - 2.292481250360578 = 5 \][/tex]
So, the value of the expression [tex]\( 3 \log_2 4 + \frac{1}{2} \log_2 6 - \frac{1}{2} \log_2 24 \)[/tex] is:
[tex]\[
\boxed{5}
\][/tex]
