The Proof For The Power Property Of Logarithms Appears In The Table With An Expression Missing. \[ \begin{tabular}{|c|c|} \hline \textbf{Step} & \textbf{Reason} \\ \hline \log _3(\sqrt{12})$ & Given \ \hline \text{[Missing Expression]} &

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, $\log_a(b^c) = c \log_a(b)$. 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 can be proven using the definition of logarithms. Let's start with the equation $\log_a(b^c) = x$. We can rewrite this equation as $a^x = b^c$. Taking the logarithm of both sides with base $a$, we get $\log_a(a^x) = \log_a(b^c)$. Using the property of logarithms that states $\log_a(a^x) = x$, we can simplify the left-hand side of the equation to $x$. Therefore, we have $x = \log_a(b^c)$.

Now, let's use the property of logarithms that states $\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$. We can rewrite the equation $\log_a(b^c) = x$ as $\frac{\log_c(b^c)}{\log_c(a)} = x$. Using the property of logarithms that states $\log_c(b^c) = c \log_c(b)$, we can simplify the numerator of the fraction to $c \log_c(b)$. Therefore, we have $\frac{c \log_c(b)}{\log_c(a)} = x$.

The Table with a Missing Expression

The table below shows the steps to prove the power property of logarithms, with a missing expression.

Step	Reason
$\log_3(\sqrt{12})$	Given
[Missing Expression]	Discussion category: mathematics

The Missing Expression

The missing expression in the table is $\log_3(12^{1/2})$. To find this expression, we can use the property of logarithms that states $\log_a(b^c) = c \log_a(b)$. In this case, we have $\log_3(12^{1/2}) = \frac{1}{2} \log_3(12)$.

Simplifying the Expression

To simplify the expression $\frac{1}{2} \log_3(12)$, we can use the property of logarithms that states $\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$. We can rewrite the expression as $\frac{1}{2} \frac{\log_c(12)}{\log_c(3)}$. Using the property of logarithms that states $\log_c(12) = \log_c(3^2 \cdot 2)$, we can simplify the numerator of the fraction to $2 \log_c(3) + \log_c(2)$.

The Final Expression

The final expression is $\frac{1}{2} \frac{2 \log_c(3) + \log_c(2)}{\log_c(3)}$. We can simplify this expression by canceling out the common factor of $\log_c(3)$ in the numerator and denominator. Therefore, we have $\frac{2 \log_c(3) + \log_c(2)}{2 \log_c(3)}$.

Conclusion

In conclusion, the power property of logarithms is a fundamental concept in mathematics that states $\log_a(b^c) = c \log_a(b)$. We have examined a table with a missing expression and found the final expression to be $\frac{2 \log_c(3) + \log_c(2)}{2 \log_c(3)}$. This expression can be simplified further by canceling out the common factor of $\log_c(3)$ in the numerator and denominator.

The Importance of the Power Property of Logarithms

The power property of logarithms is widely used in various mathematical applications, including solving equations, simplifying expressions, and modeling real-world problems. It is a fundamental concept in algebra and calculus, and is used to solve problems in physics, engineering, and computer science.

Real-World Applications

The power property of logarithms has many real-world applications. For example, it is used to solve problems in physics, such as calculating the energy of a system. It is also used in engineering to design and optimize systems. In computer science, it is used to develop algorithms and data structures.

Conclusion

In conclusion, the power property of logarithms is a fundamental concept in mathematics that states $\log_a(b^c) = c \log_a(b)$. We have examined a table with a missing expression and found the final expression to be $\frac{2 \log_c(3) + \log_c(2)}{2 \log_c(3)}$. This expression can be simplified further by canceling out the common factor of $\log_c(3)$ in the numerator and denominator. The power property of logarithms is widely used in various mathematical applications, including solving equations, simplifying expressions, and modeling real-world problems.

References

• [1] "Logarithms" by Math Open Reference
• [2] "Power Property of Logarithms" by Khan Academy
• [3] "Logarithmic Functions" by Wolfram MathWorld

Further Reading

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  Q&A: The Power Property of Logarithms =====================================

Introduction

The power property of logarithms is a fundamental concept in mathematics that states $\log_a(b^c) = c \log_a(b)$. In this article, we will answer some frequently asked questions about the power property of logarithms.

Q: What is the power property of logarithms?

A: The power property of logarithms is a mathematical property that states $\log_a(b^c) = c \log_a(b)$. This property is used to simplify expressions involving logarithms.

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

A: To use the power property of logarithms, you need to identify the base, the exponent, and the logarithm. Then, you can apply the property by multiplying the exponent by the logarithm of the base.

Q: What are some examples of the power property of logarithms?

A: Here are some examples:

• $\log_2(4^3) = 3 \log_2(4)$
• $\log_5(25^2) = 2 \log_5(25)$
• $\log_3(9^4) = 4 \log_3(9)$

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

A: Yes, you can use the power property of logarithms with different bases. For example:

• $\log_2(4^3) = 3 \log_2(4)$
• $\log_5(25^2) = 2 \log_5(25)$
• $\log_3(9^4) = 4 \log_3(9)$

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

A: Here are some common mistakes to avoid:

• Not identifying the base, exponent, and logarithm correctly
• Not applying the property correctly
• Not simplifying the expression correctly

Q: How do I simplify expressions involving logarithms using the power property?

A: To simplify expressions involving logarithms using the power property, you need to:

1. Identify the base, exponent, and logarithm
2. Apply the power property by multiplying the exponent by the logarithm of the base
3. Simplify the expression correctly

Q: Can I use the power property of logarithms with negative exponents?

A: Yes, you can use the power property of logarithms with negative exponents. For example:

• $\log_2(4^{-3}) = -3 \log_2(4)$
• $\log_5(25^{-2}) = -2 \log_5(25)$
• $\log_3(9^{-4}) = -4 \log_3(9)$

Q: How do I use the power property of logarithms with logarithms of different bases?

A: To use the power property of logarithms with logarithms of different bases, you need to:

1. Identify the base, exponent, and logarithm
2. Apply the power property by multiplying the exponent by the logarithm of the base
3. Simplify the expression correctly using the change of base formula

Conclusion

In conclusion, the power property of logarithms is a fundamental concept in mathematics that states $\log_a(b^c) = c \log_a(b)$. We have answered some frequently asked questions about the power property of logarithms, including how to use it, examples, and common mistakes to avoid. We hope this article has been helpful in understanding the power property of logarithms.

References

• [1] "Logarithms" by Math Open Reference
• [2] "Power Property of Logarithms" by Khan Academy
• [3] "Logarithmic Functions" by Wolfram MathWorld

Further Reading

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton