https://r2.erweima.ai/mathbot/image/1725675107176kroerlj3.jpg

The expression in the image is:

\[
\log(0.001) - \log(1.333) + \frac{\log(71)}{\log_4(3.73)} + 2 \log_2\left(3 \left(\frac{32}{5}\right)\right)
\]

### Step-by-step solution:

1. **Simplifying each logarithmic expression:**

   - \(\log(0.001)\):
     \[
     \log(0.001) = \log(10^{-3}) = -3
     \]

   - \(\log(1.333)\): This value can be approximated using a calculator:
     \[
     \log(1.333) \approx 0.125
     \]

   - \(\frac{\log(71)}{\log_4(3.73)}\): 

     - First, find \(\log(71)\):
       \[
       \log(71) \approx 1.851
       \]

     - Now, for \(\log_4(3.73)\), use the change of base formula:
       \[
       \log_4(3.73) = \frac{\log(3.73)}{\log(4)}
       \]

       Approximating:
       \[
       \log(3.73) \approx 0.572 \quad \text{and} \quad \log(4) = 0.602
       \]
       So:
       \[
       \log_4(3.73) \approx \frac{0.572}{0.602} \approx 0.950
       \]

     Now, we can divide:
     \[
     \frac{\log(71)}{\log_4(3.73)} \approx \frac{1.851}{0.950} \approx 1.949
     \]

   - \(2 \log_2\left(3 \left(\frac{32}{5}\right)\right)\): 

     Simplifying the expression inside the logarithm:
     \[
     3 \left(\frac{32}{5}\right) = \frac{96}{5} \approx 19.2
     \]

     Now:
     \[
     \log_2(19.2) \approx 4.263
     \]
     Therefore:
     \[
     2 \log_2(19.2) = 2 \times 4.263 \approx 8.526
     \]

2. **Putting everything together:**
   \[
   -3 - 0.125 + 1.949 + 8.526
   \]

   Combine all terms:
   \[
   -3 - 0.125 + 1.949 + 8.526 \approx 7.350
   \]

### Final answer:
\[
\approx 7.35
\]

Let me know if you'd like further details on any step, or if you have any questions! 

Here are 5 related questions:
1. How can we simplify expressions involving logarithms of different bases?
2. What is the change of base formula for logarithms, and why is it useful?
3. Can you explain how to approximate logarithms using a calculator?
4. How do we handle negative logarithmic values?
5. What is the relationship between exponential and logarithmic functions?

**Tip:** The change of base formula \(\log_b(x) = \frac{\log(x)}{\log(b)}\) is very useful when you're working with logarithms that don't match a calculator's base, such as base 2 or base 4.

log 0.001 - log 1.333 + (log 71 / log_4 3.73) + 2 log_2 (3 * (32/5))

Simplify the logarithmic expression: log 0.001 - log 1.333 + (log 71 / log_4 3.73) + 2 log_2 (3 * (32/5)). This problem covers key logarithmic concepts, including simplification techniques, change of base formula, and logarithmic properties. Suitable for high school students in Grades 10-12.

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