Correct The Mistake If Possible.a) Evaluate: \log_2\left(\frac{1}{16}\right ] - Hint: 2 ? = 1 16 2^? = \frac{1}{16} 2 ? = 16 1 ​ B) So: A) Answer: -4

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Introduction

In mathematics, logarithms are a fundamental concept that plays a crucial role in various mathematical operations. However, evaluating logarithms can be a challenging task, especially when dealing with complex expressions. In this article, we will evaluate the expression log⁡2(116)\log_2\left(\frac{1}{16}\right) and discuss the correct approach to solving it.

Evaluating the Expression

To evaluate the expression log⁡2(116)\log_2\left(\frac{1}{16}\right), we need to understand the properties of logarithms. The logarithm of a number to a certain base is the exponent to which the base must be raised to produce the number. In this case, we are looking for the exponent to which 2 must be raised to produce 116\frac{1}{16}.

Using the Hint

The given hint is 2?=1162^? = \frac{1}{16}. This hint suggests that we need to find the exponent to which 2 must be raised to produce 116\frac{1}{16}. To do this, we can rewrite 116\frac{1}{16} as a power of 2.

Rewriting 116\frac{1}{16} as a Power of 2

116\frac{1}{16} can be rewritten as 124\frac{1}{2^4}, since 24=162^4 = 16. Therefore, we can rewrite the expression as log⁡2(124)\log_2\left(\frac{1}{2^4}\right).

Applying the Property of Logarithms

One of the properties of logarithms is that log⁡b(1x)=−log⁡b(x)\log_b\left(\frac{1}{x}\right) = -\log_b(x). Using this property, we can rewrite the expression as −log⁡2(24)-\log_2(2^4).

Evaluating the Expression

Now, we can evaluate the expression −log⁡2(24)-\log_2(2^4). Since log⁡b(bx)=x\log_b(b^x) = x, we can simplify the expression to −4-4.

Conclusion

In conclusion, the correct evaluation of the expression log⁡2(116)\log_2\left(\frac{1}{16}\right) is −4-4. This result is obtained by using the properties of logarithms and rewriting the expression as a power of 2.

Common Mistakes to Avoid

When evaluating logarithmic expressions, it's essential to avoid common mistakes. Some of the common mistakes to avoid include:

  • Not using the properties of logarithms correctly
  • Not rewriting the expression as a power of the base
  • Not applying the correct property of logarithms

Tips for Evaluating Logarithmic Expressions

To evaluate logarithmic expressions correctly, follow these tips:

  • Use the properties of logarithms to simplify the expression
  • Rewrite the expression as a power of the base
  • Apply the correct property of logarithms
  • Check your work to ensure that the result is correct

Final Thoughts

Evaluating logarithmic expressions can be a challenging task, but with the correct approach and understanding of the properties of logarithms, it can be done with ease. By following the tips and avoiding common mistakes, you can ensure that your results are accurate and reliable.

Frequently Asked Questions

Q: What is the correct evaluation of the expression log⁡2(116)\log_2\left(\frac{1}{16}\right)?

A: The correct evaluation of the expression log⁡2(116)\log_2\left(\frac{1}{16}\right) is −4-4.

Q: How do I rewrite 116\frac{1}{16} as a power of 2?

A: You can rewrite 116\frac{1}{16} as 124\frac{1}{2^4}, since 24=162^4 = 16.

Q: What is the property of logarithms that I can use to simplify the expression?

A: You can use the property log⁡b(1x)=−log⁡b(x)\log_b\left(\frac{1}{x}\right) = -\log_b(x) to simplify the expression.

Q: How do I apply the correct property of logarithms?

A: You can apply the correct property of logarithms by rewriting the expression as a power of the base and then using the property to simplify the expression.

References

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Introduction

Logarithmic expressions can be a challenging topic for many students. However, with the right approach and understanding of the properties of logarithms, evaluating logarithmic expressions can be done with ease. In this article, we will provide a comprehensive Q&A guide to help you understand logarithmic expressions and how to evaluate them correctly.

Q&A Guide

Q: What is a logarithmic expression?

A: A logarithmic expression is an expression that involves a logarithm, which is the inverse operation of exponentiation. In other words, it is an expression that asks "to what power must a base be raised to produce a given number?"

Q: What are the properties of logarithms?

A: The properties of logarithms are:

  • log⁡b(1x)=−log⁡b(x)\log_b\left(\frac{1}{x}\right) = -\log_b(x)
  • log⁡b(xy)=ylog⁡b(x)\log_b(x^y) = y\log_b(x)
  • log⁡b(x)+log⁡b(y)=log⁡b(xy)\log_b(x) + \log_b(y) = \log_b(xy)
  • log⁡b(x)−log⁡b(y)=log⁡b(xy)\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you need to follow these steps:

  1. Simplify the expression using the properties of logarithms.
  2. Rewrite the expression as a power of the base.
  3. Apply the correct property of logarithms to simplify the expression.
  4. Check your work to ensure that the result is correct.

Q: What is the difference between a logarithmic expression and an exponential expression?

A: A logarithmic expression is an expression that involves a logarithm, while an exponential expression is an expression that involves an exponent. In other words, a logarithmic expression asks "to what power must a base be raised to produce a given number?", while an exponential expression asks "what number must be raised to a certain power to produce a given number?"

Q: How do I rewrite a logarithmic expression as an exponential expression?

A: To rewrite a logarithmic expression as an exponential expression, you need to follow these steps:

  1. Rewrite the expression as a power of the base.
  2. Apply the property log⁡b(x)=y\log_b(x) = y to rewrite the expression as an exponential expression.
  3. Simplify the expression to obtain the final result.

Q: What are some common mistakes to avoid when evaluating logarithmic expressions?

A: Some common mistakes to avoid when evaluating logarithmic expressions include:

  • Not using the properties of logarithms correctly
  • Not rewriting the expression as a power of the base
  • Not applying the correct property of logarithms
  • Not checking your work to ensure that the result is correct

Tips for Evaluating Logarithmic Expressions

To evaluate logarithmic expressions correctly, follow these tips:

  • Use the properties of logarithms to simplify the expression
  • Rewrite the expression as a power of the base
  • Apply the correct property of logarithms
  • Check your work to ensure that the result is correct
  • Practice, practice, practice! The more you practice evaluating logarithmic expressions, the more comfortable you will become with the process.

Frequently Asked Questions

Q: What is the correct evaluation of the expression log⁡2(116)\log_2\left(\frac{1}{16}\right)?

A: The correct evaluation of the expression log⁡2(116)\log_2\left(\frac{1}{16}\right) is −4-4.

Q: How do I rewrite 116\frac{1}{16} as a power of 2?

A: You can rewrite 116\frac{1}{16} as 124\frac{1}{2^4}, since 24=162^4 = 16.

Q: What is the property of logarithms that I can use to simplify the expression?

A: You can use the property log⁡b(1x)=−log⁡b(x)\log_b\left(\frac{1}{x}\right) = -\log_b(x) to simplify the expression.

Q: How do I apply the correct property of logarithms?

A: You can apply the correct property of logarithms by rewriting the expression as a power of the base and then using the property to simplify the expression.

References

Conclusion

Evaluating logarithmic expressions can be a challenging task, but with the right approach and understanding of the properties of logarithms, it can be done with ease. By following the tips and avoiding common mistakes, you can ensure that your results are accurate and reliable. Remember to practice, practice, practice! The more you practice evaluating logarithmic expressions, the more comfortable you will become with the process.