Evaluate Ln ⁡ ( 210 \ln (210 Ln ( 210 ]. Give Your Answer To 4 Decimal Places.

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Introduction

In this article, we will delve into the world of mathematics, specifically focusing on the natural logarithm function. We will evaluate the expression ln⁡(210)\ln (210) and provide the result to 4 decimal places. The natural logarithm function, denoted by ln⁡x\ln x, is the inverse of the exponential function exe^x. It is a fundamental concept in mathematics, with numerous applications in various fields, including physics, engineering, and economics.

The Natural Logarithm Function

The natural logarithm function is defined as the inverse of the exponential function exe^x. In other words, ln⁡x\ln x is the power to which the base ee must be raised to produce the number xx. The natural logarithm function has several important properties, including:

  • Domain and Range: The domain of the natural logarithm function is all positive real numbers, while the range is all real numbers.
  • One-to-One: The natural logarithm function is one-to-one, meaning that each output value corresponds to exactly one input value.
  • Continuous: The natural logarithm function is continuous, meaning that it can be drawn without lifting the pencil from the paper.

Evaluating ln⁡(210)\ln (210)

To evaluate ln⁡(210)\ln (210), we can use the definition of the natural logarithm function. We can rewrite the expression as:

ln⁡(210)=ln⁡(2⋅105)=ln⁡2+ln⁡105\ln (210) = \ln (2 \cdot 105) = \ln 2 + \ln 105

Using the property of logarithms that states ln⁡(ab)=ln⁡a+ln⁡b\ln (ab) = \ln a + \ln b, we can simplify the expression further:

ln⁡(210)=ln⁡2+ln⁡105=ln⁡2+ln⁡(3⋅35)=ln⁡2+ln⁡3+ln⁡35\ln (210) = \ln 2 + \ln 105 = \ln 2 + \ln (3 \cdot 35) = \ln 2 + \ln 3 + \ln 35

Now, we can use the property of logarithms that states ln⁡(ab)=ln⁡a+ln⁡b\ln (ab) = \ln a + \ln b again to simplify the expression:

ln⁡(210)=ln⁡2+ln⁡3+ln⁡35=ln⁡2+ln⁡3+ln⁡(5⋅7)=ln⁡2+ln⁡3+ln⁡5+ln⁡7\ln (210) = \ln 2 + \ln 3 + \ln 35 = \ln 2 + \ln 3 + \ln (5 \cdot 7) = \ln 2 + \ln 3 + \ln 5 + \ln 7

Using a Calculator to Evaluate ln⁡(210)\ln (210)

While we can simplify the expression using the properties of logarithms, it is often more convenient to use a calculator to evaluate the expression. Most calculators have a built-in function for the natural logarithm, which can be used to evaluate the expression.

Using a calculator, we can evaluate the expression ln⁡(210)\ln (210) as follows:

ln⁡(210)≈5.2984\ln (210) \approx 5.2984

Conclusion

In this article, we evaluated the expression ln⁡(210)\ln (210) using the properties of logarithms and a calculator. We found that the result is approximately 5.29845.2984 to 4 decimal places. The natural logarithm function is a fundamental concept in mathematics, with numerous applications in various fields. Understanding the properties and behavior of the natural logarithm function is essential for solving problems in mathematics and other disciplines.

References

Further Reading

Tags

  • Natural Logarithm
  • Logarithmic Properties
  • Calculator
  • Mathematics
  • Calculus
  • Logarithmic Functions

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Q&A: Evaluating ln⁡(210)\ln (210)

Q: What is the natural logarithm function?

A: The natural logarithm function, denoted by ln⁡x\ln x, is the inverse of the exponential function exe^x. It is a fundamental concept in mathematics, with numerous applications in various fields, including physics, engineering, and economics.

Q: How do I evaluate ln⁡(210)\ln (210)?

A: To evaluate ln⁡(210)\ln (210), you can use the definition of the natural logarithm function. You can rewrite the expression as:

ln⁡(210)=ln⁡(2⋅105)=ln⁡2+ln⁡105\ln (210) = \ln (2 \cdot 105) = \ln 2 + \ln 105

Using the property of logarithms that states ln⁡(ab)=ln⁡a+ln⁡b\ln (ab) = \ln a + \ln b, you can simplify the expression further:

ln⁡(210)=ln⁡2+ln⁡105=ln⁡2+ln⁡(3⋅35)=ln⁡2+ln⁡3+ln⁡35\ln (210) = \ln 2 + \ln 105 = \ln 2 + \ln (3 \cdot 35) = \ln 2 + \ln 3 + \ln 35

Now, you can use the property of logarithms that states ln⁡(ab)=ln⁡a+ln⁡b\ln (ab) = \ln a + \ln b again to simplify the expression:

ln⁡(210)=ln⁡2+ln⁡3+ln⁡35=ln⁡2+ln⁡3+ln⁡(5⋅7)=ln⁡2+ln⁡3+ln⁡5+ln⁡7\ln (210) = \ln 2 + \ln 3 + \ln 35 = \ln 2 + \ln 3 + \ln (5 \cdot 7) = \ln 2 + \ln 3 + \ln 5 + \ln 7

Q: Can I use a calculator to evaluate ln⁡(210)\ln (210)?

A: Yes, you can use a calculator to evaluate the expression ln⁡(210)\ln (210). Most calculators have a built-in function for the natural logarithm, which can be used to evaluate the expression.

Using a calculator, you can evaluate the expression ln⁡(210)\ln (210) as follows:

ln⁡(210)≈5.2984\ln (210) \approx 5.2984

Q: What are some common applications of the natural logarithm function?

A: The natural logarithm function has numerous applications in various fields, including:

  • Physics: The natural logarithm function is used to describe the behavior of physical systems, such as the decay of radioactive materials.
  • Engineering: The natural logarithm function is used to design and analyze complex systems, such as electronic circuits and mechanical systems.
  • Economics: The natural logarithm function is used to model economic systems, such as the behavior of stock prices and the growth of economies.

Q: What are some common properties of the natural logarithm function?

A: The natural logarithm function has several important properties, including:

  • Domain and Range: The domain of the natural logarithm function is all positive real numbers, while the range is all real numbers.
  • One-to-One: The natural logarithm function is one-to-one, meaning that each output value corresponds to exactly one input value.
  • Continuous: The natural logarithm function is continuous, meaning that it can be drawn without lifting the pencil from the paper.

Q: How do I use the natural logarithm function in real-world applications?

A: The natural logarithm function can be used in a variety of real-world applications, including:

  • Modeling population growth: The natural logarithm function can be used to model the growth of populations over time.
  • Analyzing financial data: The natural logarithm function can be used to analyze financial data, such as stock prices and interest rates.
  • Designing electronic circuits: The natural logarithm function can be used to design and analyze electronic circuits.

Conclusion

In this article, we evaluated the expression ln⁡(210)\ln (210) using the properties of logarithms and a calculator. We found that the result is approximately 5.29845.2984 to 4 decimal places. The natural logarithm function is a fundamental concept in mathematics, with numerous applications in various fields. Understanding the properties and behavior of the natural logarithm function is essential for solving problems in mathematics and other disciplines.

References

Further Reading

Tags

  • Natural Logarithm
  • Logarithmic Properties
  • Calculator
  • Mathematics
  • Calculus
  • Logarithmic Functions