Rewriting Log₁₅ 2³ Using The Power Property Of Logarithms

At the heart of logarithmic operations lies a powerful tool known as the power property of logarithms. This property allows us to manipulate logarithmic expressions in significant ways, simplifying calculations and revealing underlying relationships. In this comprehensive exploration, we'll dissect the problem of rewriting log₁₅ 2³ using the power property, providing a clear understanding of the concepts involved and demonstrating the step-by-step solution. This will not only help solve this specific problem but also equip you with the knowledge to tackle similar logarithmic challenges. Before diving into the specifics, let's first lay a solid foundation by defining what logarithms are and understanding their fundamental properties. Logarithms, in essence, are the inverse operations of exponentiation. If we have an exponential equation like b^x = y, the logarithmic form of this equation is log_b y = x. Here, 'b' is the base of the logarithm, 'y' is the argument, and 'x' is the exponent. Understanding this relationship is crucial for grasping the power property and its applications. Logarithms are not just abstract mathematical concepts; they have practical applications in various fields, including science, engineering, and finance. For instance, they are used to measure the intensity of earthquakes (Richter scale), the loudness of sound (decibels), and the pH of solutions in chemistry. Their ability to compress large ranges of values into a more manageable scale makes them invaluable in many scientific and technical contexts. Now, let's delve into the key properties of logarithms that form the basis for manipulating logarithmic expressions. These properties are essential tools for simplifying and solving logarithmic equations and are used extensively in various mathematical and scientific calculations. One of the most fundamental properties is the product rule, which states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this can be expressed as log_b (mn) = log_b m + log_b n. This property is particularly useful when dealing with expressions involving multiplication within the logarithm. Another essential property is the quotient rule, which is analogous to the product rule but applies to division. The quotient rule states that the logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator. This can be written as log_b (m/n) = log_b m - log_b n. This rule is invaluable when simplifying expressions involving division within the logarithm. Finally, we arrive at the power property, which is the focus of our discussion. The power property states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. In mathematical terms, this is expressed as log_b (m^p) = p log_b m. This property is incredibly powerful for simplifying expressions with exponents inside the logarithm, as it allows us to bring the exponent down as a coefficient. With these fundamental properties in mind, we can now turn our attention to the specific problem of rewriting log₁₅ 2³ using the power property. This property will allow us to transform the expression into a simpler, equivalent form, making it easier to work with in various mathematical contexts.

Applying the Power Property to log₁₅ 2³

Now, let's directly apply the power property to rewrite the given expression, log₁₅ 2³. Recall that the power property states: log_b (m^p) = p log_b m. In our case, the base 'b' is 15, the number 'm' is 2, and the power 'p' is 3. By substituting these values into the power property formula, we can rewrite log₁₅ 2³ as 3 log₁₅ 2. This transformation is the direct application of the power property, effectively moving the exponent from within the logarithm to a coefficient multiplying the logarithm. The power property is not just a mathematical trick; it has a profound impact on simplifying logarithmic expressions and making them more manageable. By transforming an exponent within a logarithm into a coefficient, we often gain the ability to combine terms, perform calculations, and solve equations more easily. In the context of solving logarithmic equations, the power property is frequently used to isolate the variable. For instance, if we have an equation like log_b (x^p) = c, we can use the power property to rewrite it as p log_b x = c. Then, by dividing both sides by 'p', we can isolate the logarithmic term, making it easier to solve for 'x'. In calculus, the power property is essential for differentiating and integrating logarithmic functions. The derivative of a logarithmic function often involves the power property, and understanding this property is crucial for mastering calculus techniques involving logarithms. Beyond mathematics, the power property has applications in various scientific and engineering fields. For example, in signal processing, logarithmic scales are used to represent signal amplitudes, and the power property helps in analyzing and manipulating these signals. In acoustics, the loudness of sound is measured in decibels, which is a logarithmic scale, and the power property is used to relate sound intensity to decibel levels. In chemistry, the pH scale, which measures the acidity or alkalinity of a solution, is also logarithmic, and the power property is used in calculations involving pH values. The power property can also be used in reverse, to rewrite a coefficient multiplying a logarithm as an exponent within the logarithm. This is particularly useful when we need to combine logarithmic terms with different coefficients. For example, if we have an expression like 2 log_b m + 3 log_b n, we can use the power property to rewrite it as log_b (m²) + log_b (n³). Then, we can use the product rule to combine the two logarithms into a single logarithm: log_b (m²n³). This technique is invaluable for simplifying and solving more complex logarithmic expressions. Understanding the power property is not just about memorizing a formula; it's about grasping the underlying concept and recognizing its versatility in various mathematical and scientific contexts. By mastering this property, you can unlock a powerful tool for simplifying and manipulating logarithmic expressions, making them easier to work with and opening doors to solving a wide range of problems. So, as we've seen, log₁₅ 2³ rewritten using the power property is 3 log₁₅ 2. This seemingly simple transformation is a testament to the power and elegance of logarithmic properties.

Identifying the Correct Option

Having applied the power property, we've successfully rewritten log₁₅ 2³ as 3 log₁₅ 2. Now, we need to identify the correct option among the given choices that matches this result. The options presented are:

A. log₁₅ 5 B. log₁₅ 6 C. 2 log₁₅ 3 D. 3 log₁₅ 2

By carefully comparing our result, 3 log₁₅ 2, with the provided options, it becomes clear that Option D is the correct answer. Options A, B, and C represent different logarithmic expressions and do not align with the result we obtained by applying the power property. Option A, log₁₅ 5, represents the logarithm of 5 to the base 15, which is a completely different value than 3 log₁₅ 2. There is no direct relationship or property that would allow us to transform log₁₅ 2³ into log₁₅ 5. Therefore, this option is incorrect.

Similarly, Option B, log₁₅ 6, represents the logarithm of 6 to the base 15. While 6 can be factored into 2 * 3, we cannot directly relate log₁₅ 6 to log₁₅ 2³ using any of the logarithmic properties. To transform log₁₅ 6, we would need to use the product rule, which would give us log₁₅ 2 + log₁₅ 3, which is not equal to 3 log₁₅ 2. Thus, Option B is also incorrect.

Option C, 2 log₁₅ 3, represents 2 times the logarithm of 3 to the base 15. This expression is also unrelated to log₁₅ 2³. There is no way to manipulate log₁₅ 2³ using the power property or any other logarithmic property to arrive at 2 log₁₅ 3. Therefore, Option C is not the correct answer.

Finally, Option D, 3 log₁₅ 2, perfectly matches the result we obtained by applying the power property to log₁₅ 2³. We transformed log₁₅ 2³ into 3 log₁₅ 2, making Option D the correct choice. This process of elimination and direct comparison highlights the importance of carefully applying the logarithmic properties and accurately identifying the equivalent expression. It's not enough to just know the properties; you must also be able to apply them correctly and recognize the resulting expression among a set of options. In this case, the power property provided the key to unlocking the solution, and a careful comparison of the result with the options led us to the correct answer. This exercise demonstrates the importance of not only understanding the properties of logarithms but also practicing their application to various problems. By working through examples like this, you can build your confidence and proficiency in manipulating logarithmic expressions and solving logarithmic equations. This skill is essential for success in mathematics and related fields, where logarithms play a crucial role in various applications. So, by correctly identifying Option D as the answer, we've not only solved this specific problem but also reinforced our understanding of the power property and its application in rewriting logarithmic expressions.

Conclusion: Mastering Logarithmic Properties

In conclusion, the problem of rewriting log₁₅ 2³ using the power property serves as an excellent illustration of the importance and utility of logarithmic properties. We've successfully demonstrated how the power property, which states that log_b (m^p) = p log_b m, allows us to transform logarithmic expressions by moving exponents from within the logarithm to coefficients multiplying the logarithm. This transformation not only simplifies the expression but also makes it easier to work with in various mathematical contexts. By applying the power property to log₁₅ 2³, we obtained the equivalent expression 3 log₁₅ 2. This result directly matched Option D among the given choices, confirming it as the correct answer. The other options, log₁₅ 5, log₁₅ 6, and 2 log₁₅ 3, were shown to be incorrect through careful comparison and understanding of logarithmic properties. This exercise highlights the crucial role that logarithmic properties play in simplifying and manipulating logarithmic expressions. These properties are not just abstract mathematical rules; they are powerful tools that allow us to solve equations, combine terms, and perform calculations involving logarithms with greater ease and efficiency. Mastering these properties is essential for anyone working with logarithms in mathematics, science, engineering, or any other field where logarithmic scales are used.

The power property, in particular, is a versatile tool that can be used in various ways. As we've seen, it can be used to rewrite an expression with an exponent inside the logarithm as a coefficient multiplying the logarithm. However, it can also be used in reverse, to rewrite a coefficient multiplying a logarithm as an exponent within the logarithm. This flexibility makes the power property invaluable for simplifying and solving complex logarithmic expressions. Beyond the power property, other logarithmic properties, such as the product rule and the quotient rule, are equally important. The product rule allows us to rewrite the logarithm of a product as the sum of the logarithms of the individual factors, while the quotient rule allows us to rewrite the logarithm of a quotient as the difference between the logarithms of the numerator and the denominator. These properties, along with the power property, form a comprehensive toolkit for manipulating logarithmic expressions. Understanding and applying these properties correctly is crucial for solving logarithmic equations, simplifying complex expressions, and working with logarithmic scales in various applications.

The ability to work with logarithms is a fundamental skill in many areas of mathematics and science. Logarithms are used to model a wide range of phenomena, from the growth of populations to the decay of radioactive substances. They are also used in various measurement scales, such as the Richter scale for earthquakes and the decibel scale for sound intensity. By mastering the properties of logarithms, you can unlock a powerful tool for understanding and analyzing these phenomena. In addition to their practical applications, logarithms also play a crucial role in theoretical mathematics. They are closely related to exponential functions and are used in various areas of calculus, differential equations, and complex analysis. A solid understanding of logarithms is essential for success in these advanced mathematical topics. Therefore, the effort spent mastering logarithmic properties is a worthwhile investment that will pay dividends in your mathematical and scientific pursuits. By practicing with examples, working through problems, and seeking clarification when needed, you can build a strong foundation in logarithms and unlock their full potential. This problem of rewriting log₁₅ 2³ using the power property is just one small step in the journey of mastering logarithms. By continuing to explore and practice, you can develop a deep understanding of these powerful mathematical tools and their applications.

Final Answer: The final answer is D.