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The value of the definite integral \(\int_1^e \frac{2 \ln 10 \cdot \log_{10} x}{x} \, dx\) is approximately \(0.999999\), which rounds to 1.

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Here are some related questions:
1. What is the relationship between natural logarithms and logarithms of different bases?
2. How can you derive the integral formula for logarithmic functions?
3. Why is \(\ln(10)\) involved in logarithmic changes of base?
4. What are common techniques for evaluating integrals involving logarithmic functions?
5. How would this result change if the limits of integration were different?

**Tip:** For logarithms of any base \(b\), you can convert it to the natural logarithm using the formula \(\log_b x = \frac{\ln x}{\ln b}\).

Evaluate the definite integral \(\int_1^e \frac{2 \ln 10 \cdot \log_{10} x}{x} \, dx\).

Learn how to solve the definite integral \(\int_1^e \frac{2 \ln 10 \cdot \log_{10} x}{x} \, dx\), involving logarithms and natural logarithms. This problem applies the logarithmic change of base formula and techniques from calculus, suitable for advanced high school or undergraduate students.

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