Let's solve the given expression step-by-step:
[tex]\[
\frac{1}{7} \left[ \log(57.7) - 3 \log(9.24) + 4 \log(36.6) - 2 \log(23.3) \right]
\][/tex]
### Step 1: Calculate the logarithms of each individual number
We have the logarithms already:
[tex]\[
\log(57.7) \approx 1.7612
\][/tex]
[tex]\[
\log(9.24) \approx 0.9657
\][/tex]
[tex]\[
\log(36.6) \approx 1.5635
\][/tex]
[tex]\[
\log(23.3) \approx 1.3674
\][/tex]
### Step 2: Apply the coefficients to the logarithms
Now, we need to apply the coefficients to each logarithm:
[tex]\[
\log(57.7) \approx 1.7612
\][/tex]
[tex]\[
-3 \log(9.24) \approx -3 \times 0.9657 = -2.8971
\][/tex]
[tex]\[
4 \log(36.6) \approx 4 \times 1.5635 = 6.2540
\][/tex]
[tex]\[
-2 \log(23.3) \approx -2 \times 1.3674 = -2.7348
\][/tex]
### Step 3: Combine the results
Add and subtract the logarithmic values as indicated:
[tex]\[
1.7612 - 2.8971 + 6.2540 - 2.7348
\][/tex]
Breaking it down step-by-step:
[tex]\[
1.7612 - 2.8971 = -1.1359
\][/tex]
[tex]\[
-1.1359 + 6.2540 = 5.1181
\][/tex]
[tex]\[
5.1181 - 2.7348 = 2.3833
\][/tex]
### Step 4: Apply the outer coefficient
Finally, multiply the result by [tex]\( \frac{1}{7} \)[/tex]:
[tex]\[
\frac{1}{7} \times 2.3833 \approx 0.3405
\][/tex]
### Step 5: Conclusion
So, the value of the expression is:
[tex]\[
\boxed{0.3405}
\][/tex]
