The Proof For The Power Property Of Logarithms Appears In The Table With An Expression Missing.$\[ \begin{tabular}{|l|l|} \hline \multicolumn{1}{|c|}{ Step } & \multicolumn{1}{c|}{ Reason } \\ \hline $\log _8\left(M M^r\right)$ & Given

Introduction

The power property of logarithms is a fundamental concept in mathematics, particularly in algebra and calculus. It states that the logarithm of a number raised to a power is equal to the exponent multiplied by the logarithm of the number. In other words, $\log_b (x^r) = r \log_b x$. This property is widely used in various mathematical applications, including solving equations, simplifying expressions, and modeling real-world problems. In this article, we will explore the proof for the power property of logarithms and examine a table with an expression missing.

The Power Property of Logarithms

The power property of logarithms is a direct consequence of the definition of logarithms. A logarithm is the inverse operation of exponentiation, and it is defined as the exponent to which a base must be raised to produce a given number. In other words, if $y = \log_b x$, then $b^y = x$. Using this definition, we can derive the power property of logarithms.

Derivation of the Power Property

Let's start with the definition of logarithms: $\log_b x = y \iff b^y = x$. Now, let's consider the expression $\log_b (x^r)$. We can rewrite this expression as $\log_b (x^r) = y \iff b^y = x^r$. Using the definition of logarithms, we can rewrite $b^y$ as $x$. Therefore, we have $x = x^r$. This equation is true if and only if $r = 1$. However, we can also consider the case where $r \neq 1$. In this case, we can rewrite $x^r$ as $(b^y)^r$. Using the property of exponents, we can rewrite this expression as $b^{yr}$. Therefore, we have $b^{yr} = x^r$. Taking the logarithm of both sides, we get $yr = \log_b (x^r)$. Since $y = \log_b x$, we can rewrite this equation as $r \log_b x = \log_b (x^r)$. This is the power property of logarithms.

The Table with a Missing Expression

Now, let's examine the table with an expression missing.

The table appears to be incomplete, as it only shows the expression $\log_8 (M M^r)$ without providing any further information. However, we can use the power property of logarithms to fill in the missing expression.

Filling in the Missing Expression

Using the power property of logarithms, we can rewrite the expression $\log_8 (M M^r)$ as $\log_8 (M) + \log_8 (M^r)$. Since $\log_8 (M^r) = r \log_8 (M)$, we can rewrite this expression as $\log_8 (M) + r \log_8 (M)$. Combining the terms, we get $(1 + r) \log_8 (M)$.

Conclusion

In this article, we have explored the proof for the power property of logarithms and examined a table with an expression missing. Using the definition of logarithms and the property of exponents, we have derived the power property of logarithms. We have also used this property to fill in the missing expression in the table. The power property of logarithms is a fundamental concept in mathematics, and it has numerous applications in various fields. Understanding this property is essential for solving equations, simplifying expressions, and modeling real-world problems.

Applications of the Power Property of Logarithms

The power property of logarithms has numerous applications in various fields, including algebra, calculus, and computer science. Some of the applications of this property include:

• Solving equations: The power property of logarithms can be used to solve equations involving logarithms. For example, if we have the equation $\log_b x = 2$, we can rewrite it as $b^2 = x$.
• Simplifying expressions: The power property of logarithms can be used to simplify expressions involving logarithms. For example, if we have the expression $\log_b (x^r)$, we can rewrite it as $r \log_b x$.
• Modeling real-world problems: The power property of logarithms can be used to model real-world problems involving growth and decay. For example, if we have a population that grows at a rate of $r$% per year, we can use the power property of logarithms to model the population growth.

Examples of the Power Property of Logarithms

Here are some examples of the power property of logarithms:

• Example 1: Find the value of $\log_2 (4^3)$.
• Solution: Using the power property of logarithms, we can rewrite this expression as $3 \log_2 4$. Since $\log_2 4 = 2$, we have $3 \log_2 4 = 3 \cdot 2 = 6$.
• Example 2: Find the value of $\log_5 (25^2)$.
• Solution: Using the power property of logarithms, we can rewrite this expression as $2 \log_5 25$. Since $\log_5 25 = 2$, we have $2 \log_5 25 = 2 \cdot 2 = 4$.

Conclusion

In conclusion, the power property of logarithms is a fundamental concept in mathematics that has numerous applications in various fields. Understanding this property is essential for solving equations, simplifying expressions, and modeling real-world problems. We have explored the proof for the power property of logarithms and examined a table with an expression missing. Using the definition of logarithms and the property of exponents, we have derived the power property of logarithms and filled in the missing expression in the table.

Introduction

The power property of logarithms is a fundamental concept in mathematics that has numerous applications in various fields. However, it can be challenging to understand and apply this property, especially for students and professionals who are new to logarithms. In this article, we will answer some frequently asked questions (FAQs) about the power property of logarithms, providing a comprehensive guide to this important mathematical concept.

Q1: What is the power property of logarithms?

A1: The power property of logarithms states that the logarithm of a number raised to a power is equal to the exponent multiplied by the logarithm of the number. In other words, $\log_b (x^r) = r \log_b x$.

Q2: How do I apply the power property of logarithms?

A2: To apply the power property of logarithms, you need to identify the base, the exponent, and the logarithm. Then, you can use the formula $\log_b (x^r) = r \log_b x$ to rewrite the expression.

Q3: What are some common mistakes to avoid when using the power property of logarithms?

A3: Some common mistakes to avoid when using the power property of logarithms include:

• Forgetting to multiply the exponent by the logarithm: Make sure to multiply the exponent by the logarithm to get the correct result.
• Using the wrong base: Make sure to use the correct base for the logarithm.
• Not simplifying the expression: Make sure to simplify the expression after applying the power property of logarithms.

Q4: How do I use the power property of logarithms to solve equations?

A4: To use the power property of logarithms to solve equations, you need to isolate the logarithm on one side of the equation. Then, you can apply the power property of logarithms to rewrite the expression.

Q5: Can I use the power property of logarithms to simplify expressions?

A5: Yes, you can use the power property of logarithms to simplify expressions. For example, if you have the expression $\log_b (x^r)$, you can rewrite it as $r \log_b x$ using the power property of logarithms.

Q6: What are some real-world applications of the power property of logarithms?

A6: Some real-world applications of the power property of logarithms include:

• Modeling population growth: The power property of logarithms can be used to model population growth and decay.
• Analyzing financial data: The power property of logarithms can be used to analyze financial data and make predictions about future trends.
• Solving optimization problems: The power property of logarithms can be used to solve optimization problems in fields such as engineering and economics.

Q7: Can I use the power property of logarithms with different bases?

A7: Yes, you can use the power property of logarithms with different bases. For example, if you have the expression $\log_2 (x^r)$, you can rewrite it as $r \log_2 x$ using the power property of logarithms.

Q8: How do I use the power property of logarithms with negative exponents?

A8: To use the power property of logarithms with negative exponents, you need to rewrite the expression with a positive exponent. Then, you can apply the power property of logarithms to rewrite the expression.

Q9: Can I use the power property of logarithms with fractional exponents?

A9: Yes, you can use the power property of logarithms with fractional exponents. For example, if you have the expression $\log_b (x^{1/2})$, you can rewrite it as $\frac{1}{2} \log_b x$ using the power property of logarithms.

Q10: How do I use the power property of logarithms with logarithmic expressions?

A10: To use the power property of logarithms with logarithmic expressions, you need to rewrite the expression using the power property of logarithms. Then, you can simplify the expression to get the final result.

Conclusion

In conclusion, the power property of logarithms is a fundamental concept in mathematics that has numerous applications in various fields. By understanding this property and how to apply it, you can solve equations, simplify expressions, and model real-world problems. We hope that this article has provided a comprehensive guide to the power property of logarithms and has helped you to better understand this important mathematical concept.