Which Expressions Are Equivalent To The One Below? Check All That Apply. \log \left(10^7\right ]A. 7 ⋅ Log ⁡ 10 7 \cdot \log 10 7 ⋅ Lo G 10 B. 7 C. 7 ⋅ 10 7 \cdot 10 7 ⋅ 10 D. 1

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In mathematics, logarithms are a fundamental concept used to solve equations and express complex relationships between numbers. The logarithmic function is the inverse of the exponential function, and it plays a crucial role in various mathematical operations. In this article, we will explore the equivalent expressions of the given logarithmic expression log⁡(107)\log \left(10^7\right) and identify the correct options.

The Given Expression

The given expression is log⁡(107)\log \left(10^7\right). This expression represents the logarithm of 10710^7 to the base 10. To understand this expression, let's break it down:

  • The base of the logarithm is 10.
  • The argument of the logarithm is 10710^7.

Understanding Logarithmic Properties

To simplify the given expression, we need to understand the properties of logarithms. The logarithmic function has several properties that help us simplify and manipulate logarithmic expressions. Some of the key properties are:

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

Simplifying the Given Expression

Using the power property of logarithms, we can simplify the given expression as follows:

log⁡(107)=7⋅log⁡10\log \left(10^7\right) = 7 \cdot \log 10

This expression represents the logarithm of 10710^7 to the base 10, which is equivalent to 7 times the logarithm of 10.

Evaluating the Options

Now that we have simplified the given expression, let's evaluate the options:

A. 7⋅log⁡107 \cdot \log 10 B. 7 C. 7⋅107 \cdot 10 D. 1

Based on our simplification, option A is the correct equivalent expression. The other options are incorrect.

Why Option A is Correct

Option A is correct because it represents the simplified expression we obtained using the power property of logarithms. The expression 7⋅log⁡107 \cdot \log 10 is equivalent to the original expression log⁡(107)\log \left(10^7\right).

Why Options B, C, and D are Incorrect

Options B, C, and D are incorrect because they do not represent the simplified expression we obtained using the power property of logarithms.

  • Option B is incorrect because it represents a numerical value (7) rather than an expression.
  • Option C is incorrect because it represents a product of 7 and 10, rather than a logarithmic expression.
  • Option D is incorrect because it represents a numerical value (1) rather than an expression.

Conclusion

In conclusion, the equivalent expressions of the given logarithmic expression log⁡(107)\log \left(10^7\right) are:

  • 7⋅log⁡107 \cdot \log 10

The other options are incorrect. This article has demonstrated the importance of understanding logarithmic properties and how to simplify logarithmic expressions using these properties.

Additional Examples

To further illustrate the concept of logarithmic equivalences, let's consider some additional examples:

  • log⁡(103)=3⋅log⁡10\log \left(10^3\right) = 3 \cdot \log 10
  • log⁡(10−2)=−2⋅log⁡10\log \left(10^{-2}\right) = -2 \cdot \log 10
  • log⁡(1012)=12⋅log⁡10\log \left(10^{\frac{1}{2}}\right) = \frac{1}{2} \cdot \log 10

These examples demonstrate how to simplify logarithmic expressions using the power property of logarithms.

Final Thoughts

In our previous article, we explored the concept of logarithmic equivalences and how to simplify logarithmic expressions using the power property of logarithms. In this article, we will provide a Q&A guide to help you better understand and apply this concept.

Q: What is the power property of logarithms?

A: The power property of logarithms states that log⁡ab=b⋅log⁡a\log a^b = b \cdot \log a. This property allows us to simplify logarithmic expressions by rewriting them in terms of the logarithm of a single number.

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

A: To apply the power property of logarithms, follow these steps:

  1. Identify the base and the exponent in the logarithmic expression.
  2. Rewrite the expression using the power property: log⁡ab=b⋅log⁡a\log a^b = b \cdot \log a.
  3. Simplify the expression by evaluating the logarithm of the base.

Q: What are some common logarithmic expressions that can be simplified using the power property?

A: Some common logarithmic expressions that can be simplified using the power property include:

  • log⁡(107)=7⋅log⁡10\log \left(10^7\right) = 7 \cdot \log 10
  • log⁡(10−2)=−2⋅log⁡10\log \left(10^{-2}\right) = -2 \cdot \log 10
  • log⁡(1012)=12⋅log⁡10\log \left(10^{\frac{1}{2}}\right) = \frac{1}{2} \cdot \log 10

Q: Can I use the power property of logarithms to simplify expressions with different bases?

A: No, the power property of logarithms only applies to expressions with the same base. If you have an expression with a different base, you will need to use a different property or technique to simplify it.

Q: How do I evaluate logarithmic expressions with negative exponents?

A: To evaluate logarithmic expressions with negative exponents, follow these steps:

  1. Rewrite the expression using the power property: log⁡a−b=−b⋅log⁡a\log a^{-b} = -b \cdot \log a.
  2. Simplify the expression by evaluating the logarithm of the base.

Q: Can I use the power property of logarithms to simplify expressions with fractional exponents?

A: Yes, you can use the power property of logarithms to simplify expressions with fractional exponents. For example:

  • log⁡(1012)=12⋅log⁡10\log \left(10^{\frac{1}{2}}\right) = \frac{1}{2} \cdot \log 10

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

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

  • Forgetting to apply the power property of logarithms.
  • Using the wrong property or technique to simplify the expression.
  • Not evaluating the logarithm of the base correctly.

Q: How do I check my work when simplifying logarithmic expressions?

A: To check your work when simplifying logarithmic expressions, follow these steps:

  1. Evaluate the logarithmic expression using the original expression.
  2. Compare the result to the simplified expression.
  3. If the results are the same, then your work is correct.

Conclusion

In conclusion, this Q&A guide has provided you with a better understanding of logarithmic equivalences and how to simplify logarithmic expressions using the power property of logarithms. By following the steps outlined in this guide, you can confidently simplify logarithmic expressions and solve equations with ease.

Additional Resources

For further practice and review, try the following exercises:

  • Simplify the following logarithmic expressions using the power property of logarithms:
    • log⁡(103)\log \left(10^3\right)
    • log⁡(10−2)\log \left(10^{-2}\right)
    • log⁡(1012)\log \left(10^{\frac{1}{2}}\right)
  • Evaluate the following logarithmic expressions using the power property of logarithms:
    • log⁡(107)\log \left(10^7\right)
    • log⁡(10−2)\log \left(10^{-2}\right)
    • log⁡(1012)\log \left(10^{\frac{1}{2}}\right)

By practicing these exercises, you will become more confident and proficient in simplifying logarithmic expressions using the power property of logarithms.