Which Expressions Are Equivalent To The One Below? Check All That Apply. Log ⁡ 5 5 + Log ⁡ 5 125 \log _5 5 + \log _5 125 Lo G 5 ​ 5 + Lo G 5 ​ 125 A. Log ⁡ 10 \log 10 Lo G 10 B. 4C. Log ⁡ 5 625 \log _5 625 Lo G 5 ​ 625 D. Log ⁡ 5 ( 5 4 \log _5(5^4 Lo G 5 ​ ( 5 4 ]

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will delve into the world of logarithms and explore the equivalent expressions of the given logarithmic expression: $\log _5 5 + \log _5 125$. We will examine each option carefully and determine which ones are equivalent to the given expression.

The Given Expression

The given expression is $\log _5 5 + \log _5 125$. To understand this expression, let's break it down into its components. The first term, $\log _5 5$, represents the logarithm of 5 to the base 5. This means that $5^x = 5$, where x is the exponent. Since $5^1 = 5$, we can conclude that $\log _5 5 = 1$.

The second term, $\log _5 125$, represents the logarithm of 125 to the base 5. To evaluate this expression, we need to find the exponent to which 5 must be raised to obtain 125. Since $5^3 = 125$, we can conclude that $\log _5 125 = 3$.

Evaluating the Given Expression

Now that we have evaluated the individual terms, let's substitute their values back into the original expression: $\log _5 5 + \log _5 125 = 1 + 3 = 4$. Therefore, the given expression is equivalent to $\log _5 625$.

Analyzing the Options

Let's examine each option carefully and determine which ones are equivalent to the given expression.

A. $\log 10$

This option is not equivalent to the given expression. The base of the logarithm is 10, whereas the base of the given expression is 5. Therefore, this option is not a correct equivalent.

B. 4

As we have already evaluated, the given expression is equivalent to 4. Therefore, this option is a correct equivalent.

C. $\log _5 625$

As we have already concluded, the given expression is equivalent to $\log _5 625$. Therefore, this option is a correct equivalent.

D. $\log _5(5^4)$

This option is not equivalent to the given expression. The expression $\log _5(5^4)$ represents the logarithm of $5^4$ to the base 5, which is equivalent to 4. However, the given expression is equivalent to $\log _5 625$, not $\log _5(5^4)$. Therefore, this option is not a correct equivalent.

Conclusion

In conclusion, the equivalent expressions of the given logarithmic expression $\log _5 5 + \log _5 125$ are $\log _5 625$ and 4. These expressions are equivalent because they both represent the same value, which is 4. Therefore, the correct options are B and C.

Key Takeaways

• Logarithmic expressions can be evaluated by breaking them down into their components.
• Understanding the properties of logarithms is crucial for solving various mathematical problems.
• The base of the logarithm is an essential component of a logarithmic expression.
• Equivalent expressions can be determined by evaluating the individual terms and substituting their values back into the original expression.

Final Thoughts

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Introduction

In our previous article, we explored the equivalent expressions of the given logarithmic expression $\log _5 5 + \log _5 125$. We have evaluated each option carefully and determined which ones are equivalent to the given expression. In this article, we will provide a comprehensive Q&A guide to help you better understand logarithmic expressions.

Q&A Guide

Q: What is a logarithmic expression?

A: A logarithmic expression is a mathematical expression that represents the power to which a base number must be raised to obtain a given value.

Q: What is the base of a logarithmic expression?

A: The base of a logarithmic expression is the number to which the power is raised to obtain the given value.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you need to break it down into its components and evaluate each term separately. Then, substitute the values back into the original expression.

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

A: A logarithmic expression represents the power to which a base number must be raised to obtain a given value, while an exponential expression represents the result of raising a base number to a given power.

Q: Can a logarithmic expression have a negative exponent?

A: Yes, a logarithmic expression can have a negative exponent. In this case, the expression is equivalent to the reciprocal of the original expression.

Q: How do I determine the equivalent expressions of a given logarithmic expression?

A: To determine the equivalent expressions of a given logarithmic expression, you need to evaluate each term separately and substitute the values back into the original expression.

Q: What is the relationship between logarithmic and exponential expressions?

A: Logarithmic and exponential expressions are inverse operations. This means that if you have a logarithmic expression, you can convert it to an exponential expression by raising the base to the power of the logarithm.

Q: Can a logarithmic expression have a fractional exponent?

A: Yes, a logarithmic expression can have a fractional exponent. In this case, the expression is equivalent to the logarithm of the base raised to the power of the fractional exponent.

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you need to combine like terms and evaluate any exponential expressions.

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 e (approximately 2.718).

Q: Can a logarithmic expression have a complex number as its base?

A: Yes, a logarithmic expression can have a complex number as its base. In this case, the expression is equivalent to the logarithm of the complex number raised to the power of the logarithm.

Conclusion

In conclusion, logarithmic expressions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. By following the Q&A guide provided in this article, you can better understand logarithmic expressions and determine their equivalent expressions.

Key Takeaways

• Logarithmic expressions represent the power to which a base number must be raised to obtain a given value.
• The base of a logarithmic expression is the number to which the power is raised to obtain the given value.
• Logarithmic and exponential expressions are inverse operations.
• A logarithmic expression can have a negative exponent, a fractional exponent, or a complex number as its base.
• To simplify a logarithmic expression, you need to combine like terms and evaluate any exponential expressions.

Final Thoughts

In this article, we have provided a comprehensive Q&A guide to help you better understand logarithmic expressions. By following the guide, you can determine the equivalent expressions of a given logarithmic expression and simplify complex logarithmic expressions.