$\log_{15} 2^3$ Rewritten Using The Power Property

Introduction

In the realm of mathematics, logarithms serve as a powerful tool for simplifying complex calculations and revealing hidden relationships between numbers. Understanding the properties of logarithms is crucial for manipulating and solving logarithmic equations effectively. Among these properties, the power property stands out as a fundamental rule that allows us to rewrite logarithmic expressions involving exponents. In this comprehensive exploration, we delve into the intricacies of logarithms, focusing specifically on the power property and its application in rewriting the expression $\log_{15} 2^3$. We will dissect the power property, illustrate its mechanics, and demonstrate how it can be used to transform logarithmic expressions into simpler, more manageable forms. By the end of this discussion, you will have a solid grasp of the power property and its role in simplifying logarithmic expressions, empowering you to tackle a wider range of mathematical problems with confidence. This foundational knowledge will not only enhance your understanding of logarithms but also lay the groundwork for exploring more advanced mathematical concepts.

Unveiling the Power Property of Logarithms

At the heart of our exploration lies the power property of logarithms, a cornerstone concept that governs how we manipulate logarithmic expressions involving exponents. This property provides a direct link between logarithms and exponents, enabling us to simplify expressions and solve equations more efficiently. The power property states that the logarithm of a number raised to an exponent is equal to the product of the exponent and the logarithm of the number. In mathematical notation, this is expressed as:

$$\log_b a^n = n \log_b a$$

where:

• b is the base of the logarithm (b > 0 and b ≠ 1)
• a is a positive real number (a > 0)
• n is any real number

The power property essentially allows us to move an exponent from inside the logarithm to the outside as a coefficient, or vice versa. This seemingly simple operation has profound implications for simplifying complex logarithmic expressions. To truly appreciate the power property, let's break down its components and explore its underlying logic. The expression $\log_b a^n$ represents the power to which we must raise the base b to obtain the value $a^n$. The power property tells us that this is the same as n times the power to which we must raise b to obtain a. In other words, we can "pull out" the exponent n and multiply it by the logarithm of a to the base b. This manipulation often simplifies calculations and makes logarithmic expressions easier to work with. The power property is not just a mathematical trick; it reflects the fundamental relationship between logarithms and exponents. It's a powerful tool that allows us to unravel the intricacies of logarithmic expressions and reveal their underlying structure. By mastering this property, we unlock a deeper understanding of logarithms and their applications in various fields, from mathematics and science to engineering and finance.

Applying the Power Property to $\log_{15} 2^3$

Now, let's put the power property into action by applying it to the specific expression $\log_{15} 2^3$. Our goal is to rewrite this expression in a simpler form using the power property. Recall that the power property states:

$$\log_b a^n = n \log_b a$$

In our case, we have:

• b = 15 (the base of the logarithm)
• a = 2 (the number inside the logarithm)
• n = 3 (the exponent)

Substituting these values into the power property formula, we get:

$$\log_{15} 2^3 = 3 \log_{15} 2$$

Therefore, by applying the power property, we have successfully rewritten $\log_{15} 2^3$ as $3 \log_{15} 2$. This transformation might seem simple, but it highlights the power of the power property in simplifying logarithmic expressions. We have effectively moved the exponent 3 from inside the logarithm to the outside as a coefficient, resulting in a more concise and manageable expression. This rewritten form is often more convenient for further calculations or manipulations. For example, if we needed to approximate the value of $\log_{15} 2^3$, we could first find the value of $\log_{15} 2$ and then multiply it by 3. This approach can be significantly easier than directly calculating $2^3$ and then finding its logarithm to the base 15. The power property is not just a theoretical concept; it has practical applications in various mathematical contexts. It allows us to break down complex logarithmic expressions into simpler components, making them easier to understand and work with. By mastering this property, we gain a valuable tool for solving logarithmic equations, simplifying calculations, and gaining deeper insights into the behavior of logarithmic functions.

Analyzing the Answer Choices

Having successfully applied the power property to rewrite $\log_{15} 2^3$ as $3 \log_{15} 2$, we now turn our attention to the given answer choices to identify the correct option. Let's examine each choice carefully and compare it with our result:

• A. $\log_{15} 5$: This option does not match our rewritten expression $3 \log_{15} 2$. It involves the logarithm of 5, which is not present in our result.
• B. $\log_{15} 6$: Similar to option A, this choice does not align with our result. It involves the logarithm of 6, which is not obtained by applying the power property to the original expression.
• C. $2 \log_{15} 3$: This option is also incorrect. While it has the form of a coefficient multiplied by a logarithm, the base and the number inside the logarithm do not match our result. We obtained $3 \log_{15} 2$, not $2 \log_{15} 3$.
• D. $3 \log_{15} 2$: This option perfectly matches our rewritten expression. We applied the power property to $\log_{15} 2^3$ and arrived at $3 \log_{15} 2$. Therefore, this is the correct answer.

By systematically analyzing each answer choice and comparing it with our derived result, we have confidently identified option D as the correct answer. This exercise highlights the importance of understanding and applying the power property accurately. It also demonstrates the value of careful analysis and comparison in problem-solving. When faced with multiple-choice questions involving logarithmic expressions, it's crucial to rewrite the expression using the appropriate properties and then compare the result with the given options. This approach ensures that you select the correct answer with confidence and avoid common pitfalls.

Conclusion: Mastering the Power Property

In this comprehensive exploration, we have delved into the power property of logarithms and its application in rewriting logarithmic expressions. We began by introducing the fundamental concept of logarithms and their significance in mathematics. We then focused on the power property, a key rule that allows us to simplify expressions involving logarithms and exponents. We dissected the property, illustrating its mechanics and underlying logic. We demonstrated how the power property can be used to move exponents from inside the logarithm to the outside as coefficients, resulting in simpler and more manageable expressions. Applying this property to the expression $\log_{15} 2^3$, we successfully rewrote it as $3 \log_{15} 2$. Furthermore, we analyzed the given answer choices and confidently identified the correct option based on our result. This journey through the power property has underscored its importance in simplifying logarithmic expressions and solving logarithmic equations. By mastering this property, you equip yourself with a valuable tool for tackling a wide range of mathematical problems. The power property is not just a mathematical trick; it's a reflection of the fundamental relationship between logarithms and exponents. It allows us to unravel the intricacies of logarithmic expressions and gain a deeper understanding of their behavior. As you continue your exploration of mathematics, remember the power property and its ability to transform complex expressions into simpler forms. Embrace this property, practice its application, and watch your problem-solving skills soar. The world of logarithms holds many fascinating secrets, and the power property is one of the keys to unlocking them.