Given That $\log (7) \approx 0.8451$, Find The Value Of The Logarithm $\log \left(\frac{1}{7000}\right$\].

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Introduction

Logarithms are a fundamental concept in mathematics, used to solve equations and express complex relationships between numbers. In this article, we will explore the concept of logarithms and how to find the value of a complex logarithm using the given approximation of log⁡(7)≈0.8451\log (7) \approx 0.8451. We will delve into the world of logarithms, discussing the properties and rules that govern their behavior.

Understanding Logarithms

A logarithm is the inverse operation of exponentiation. In other words, it is the power to which a base number must be raised to obtain a given value. For example, if we have the equation 2x=82^x = 8, then the logarithm of 8 to the base 2 is x=log⁡2(8)x = \log_2 (8). This means that 22 raised to the power of xx equals 88.

Properties of Logarithms

Logarithms have several properties that make them useful in mathematics. Some of the key properties include:

  • Product Rule: log⁡(ab)=log⁡(a)+log⁡(b)\log (ab) = \log (a) + \log (b)
  • Quotient Rule: log⁡(ab)=log⁡(a)−log⁡(b)\log \left(\frac{a}{b}\right) = \log (a) - \log (b)
  • Power Rule: log⁡(ab)=blog⁡(a)\log (a^b) = b \log (a)

These properties will be essential in finding the value of the complex logarithm log⁡(17000)\log \left(\frac{1}{7000}\right).

Finding the Value of the Complex Logarithm

To find the value of the complex logarithm log⁡(17000)\log \left(\frac{1}{7000}\right), we can use the properties of logarithms. We can rewrite the expression as:

log⁡(17000)=log⁡(1)−log⁡(7000)\log \left(\frac{1}{7000}\right) = \log (1) - \log (7000)

Using the properties of logarithms, we can simplify the expression further:

log⁡(1)=0\log (1) = 0

log⁡(7000)=log⁡(74⋅103)\log (7000) = \log (7^4 \cdot 10^3)

Using the product rule, we can rewrite the expression as:

log⁡(74⋅103)=4log⁡(7)+3log⁡(10)\log (7^4 \cdot 10^3) = 4 \log (7) + 3 \log (10)

Now, we can substitute the given approximation of log⁡(7)≈0.8451\log (7) \approx 0.8451:

4log⁡(7)+3log⁡(10)≈4(0.8451)+3log⁡(10)4 \log (7) + 3 \log (10) \approx 4(0.8451) + 3 \log (10)

Using a calculator, we can find the value of log⁡(10)\log (10):

log⁡(10)≈1\log (10) \approx 1

Substituting this value, we get:

4(0.8451)+3(1)≈3.3804+34(0.8451) + 3(1) \approx 3.3804 + 3

≈6.3804\approx 6.3804

Therefore, the value of the complex logarithm log⁡(17000)\log \left(\frac{1}{7000}\right) is approximately −6.3804-6.3804.

Conclusion

In this article, we explored the concept of logarithms and how to find the value of a complex logarithm using the given approximation of log⁡(7)≈0.8451\log (7) \approx 0.8451. We discussed the properties and rules that govern the behavior of logarithms and applied them to find the value of the complex logarithm log⁡(17000)\log \left(\frac{1}{7000}\right). The result is approximately −6.3804-6.3804. This demonstrates the power of logarithms in solving complex mathematical problems.

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Additional Resources

Introduction

Logarithms can be a complex and confusing topic, especially for those who are new to mathematics. In this article, we will answer some of the most frequently asked questions about logarithms, providing a clear and concise explanation of the concepts.

Q: What is a logarithm?

A: A logarithm is the inverse operation of exponentiation. In other words, it is the power to which a base number must be raised to obtain a given value.

Q: What is the difference between a logarithm and an exponent?

A: A logarithm and an exponent are inverse operations. For example, if we have the equation 2x=82^x = 8, then the logarithm of 8 to the base 2 is x=log⁡2(8)x = \log_2 (8). This means that 22 raised to the power of xx equals 88.

Q: What are the properties of logarithms?

A: Logarithms have several properties that make them useful in mathematics. Some of the key properties include:

  • Product Rule: log⁡(ab)=log⁡(a)+log⁡(b)\log (ab) = \log (a) + \log (b)
  • Quotient Rule: log⁡(ab)=log⁡(a)−log⁡(b)\log \left(\frac{a}{b}\right) = \log (a) - \log (b)
  • Power Rule: log⁡(ab)=blog⁡(a)\log (a^b) = b \log (a)

Q: How do I find the value of a logarithm?

A: To find the value of a logarithm, you can use the properties of logarithms. For example, if we have the equation log⁡(74⋅103)\log (7^4 \cdot 10^3), we can use the product rule to rewrite it as:

log⁡(74⋅103)=4log⁡(7)+3log⁡(10)\log (7^4 \cdot 10^3) = 4 \log (7) + 3 \log (10)

Q: What is the difference between a common logarithm and a natural logarithm?

A: A common logarithm is a logarithm with a base of 10, while a natural logarithm is a logarithm with a base of ee. The natural logarithm is often denoted as ln⁡(x)\ln (x).

Q: How do I use logarithms in real-life situations?

A: Logarithms are used in a wide range of real-life situations, including:

  • Finance: Logarithms are used to calculate interest rates and investment returns.
  • Science: Logarithms are used to calculate the pH of a solution and the concentration of a substance.
  • Engineering: Logarithms are used to calculate the power of a signal and the frequency of a wave.

Q: What are some common mistakes to avoid when working with logarithms?

A: Some common mistakes to avoid when working with logarithms include:

  • Forgetting to change the base: When working with logarithms, it is essential to change the base to the correct value.
  • Not using the correct property: Make sure to use the correct property of logarithms, such as the product rule or the quotient rule.
  • Not checking the domain: Make sure to check the domain of the logarithm to ensure that it is valid.

Conclusion

In this article, we have answered some of the most frequently asked questions about logarithms, providing a clear and concise explanation of the concepts. We hope that this article has been helpful in understanding logarithms and how to use them in real-life situations.

References

Additional Resources