Math Problem Statement

evaluate

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

Solution

The value of the definite integral ∫1e2ln⁡10⋅log⁡10xx dx\int_1^e \frac{2 \ln 10 \cdot \log_{10} x}{x} \, dx is approximately 0.9999990.999999, which rounds to 1.

Would you like further details or have any questions about this solution?

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)\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 bb, you can convert it to the natural logarithm using the formula log⁡bx=ln⁡xln⁡b\log_b x = \frac{\ln x}{\ln b}.

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Math Problem Analysis

Mathematical Concepts

Integration
Logarithms
Change of Base Formula

Formulas

Change of base formula: \(\log_b x = \frac{\ln x}{\ln b}\)
Integration formula involving logarithms: \(\int \frac{\log_b x}{x} dx\)

Theorems

Logarithmic Integration Theorem
Fundamental Theorem of Calculus

Suitable Grade Level

Undergraduate or advanced high school level (Calculus)

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