Evaluate Log₃√[4]{27} A Step-by-Step Solution
Introduction to Logarithms
In the realm of mathematics, logarithms serve as a fundamental tool for simplifying complex calculations and understanding exponential relationships. At its core, a logarithm is the inverse operation to exponentiation, much like subtraction is the inverse of addition, or division is the inverse of multiplication. This means that if we have an exponential expression, say b raised to the power of x equals y (written as bx = y), then the logarithm answers the question: To what power must we raise the base b to obtain y? This is expressed logarithmically as logb(y) = x. Understanding this fundamental relationship is key to unlocking the power of logarithms in various mathematical and scientific applications.
Logarithms are not merely abstract mathematical concepts; they have practical applications across a wide array of disciplines. For instance, in the field of acoustics, the decibel scale, which measures sound intensity, is logarithmic. This is because the human ear perceives sound intensity logarithmically, allowing us to handle a vast range of sound pressures. Similarly, in chemistry, the pH scale, which measures the acidity or alkalinity of a solution, is also logarithmic. This allows chemists to express a wide range of hydrogen ion concentrations in a manageable scale. In computer science, logarithms are crucial in analyzing the efficiency of algorithms, particularly in sorting and searching algorithms. Moreover, logarithms find applications in finance, where they are used to calculate compound interest and analyze investment growth. The ubiquity of logarithms in various fields highlights their importance in modeling and understanding phenomena that exhibit exponential growth or decay.
The concept of logarithms can be further demystified by exploring its components. The base of the logarithm, denoted as b, is the number that is raised to a power. This base is a crucial part of the logarithmic expression, as it defines the scale against which the logarithm is measured. For example, in the common logarithm (log base 10), the base is 10, while in the natural logarithm (log base e), the base is Euler's number, approximately 2.71828. The argument of the logarithm, denoted as y, is the value for which we are trying to find the logarithm. It is the result of raising the base to a certain power. The logarithm itself, denoted as x, is the exponent to which the base must be raised to obtain the argument. Understanding these components—base, argument, and logarithm—is essential for correctly interpreting and manipulating logarithmic expressions. To solidify this understanding, let's consider a simple example: log₂(8) = 3. Here, the base is 2, the argument is 8, and the logarithm is 3. This means that 2 raised to the power of 3 equals 8 (2³ = 8), thus illustrating the fundamental relationship between logarithms and exponents.
Breaking Down the Problem: log₃√[4]{27}
Now, let's tackle the specific problem at hand: log₃√[4]{27}. This expression asks us to determine the power to which we must raise the base 3 to obtain the fourth root of 27. To solve this, we'll employ a step-by-step approach, leveraging the properties of logarithms and exponents to simplify the expression. Our first step involves understanding the radical expression √[4]{27}, which represents the fourth root of 27. We can rewrite this radical using fractional exponents, a powerful technique that allows us to bridge the gap between radicals and exponents. Recall that the nth root of a number x can be expressed as x1/n. Applying this principle, we can rewrite √[4]{27} as 271/4. This transformation is crucial because it allows us to manipulate the expression using the rules of exponents, which are often easier to work with than radicals.
The second step in solving log₃√[4]{27} involves further simplifying the base within the exponent. We observe that 27 can be expressed as 3 cubed (3³). This recognition is pivotal because it allows us to express the entire argument in terms of the base of the logarithm, which is also 3. By substituting 3³ for 27 in the expression 271/4, we get (3³)1/4. Now, we can apply another fundamental rule of exponents: the power of a power rule. This rule states that when raising a power to another power, we multiply the exponents. In our case, we are raising 3³ to the power of 1/4, so we multiply the exponents 3 and 1/4, resulting in 33/4. This simplification is a significant step forward, as it has transformed the argument of the logarithm into a single exponential term with the same base as the logarithm.
With the argument simplified to 33/4, the third step in evaluating log₃√[4]{27} is to apply the fundamental definition of logarithms. Recall that logb(y) = x means that bx = y. In our case, we have log₃(33/4). Comparing this to the definition, we see that b is 3, y is 33/4, and we are trying to find x. The expression log₃(33/4) is essentially asking: To what power must we raise 3 to obtain 33/4? The answer is immediately apparent: we must raise 3 to the power of 3/4. Therefore, log₃(33/4) = 3/4. This result showcases the elegance and efficiency of logarithms in simplifying expressions involving exponents and radicals. By breaking down the problem into manageable steps and applying the properties of logarithms and exponents, we have successfully found the value of the given logarithmic expression. This process not only provides the answer but also reinforces our understanding of the underlying mathematical principles.
Applying Logarithmic Properties
To delve deeper into solving log₃√[4]{27}, we can explicitly use logarithmic properties to illustrate the solution process. Logarithmic properties are a set of rules that allow us to manipulate logarithmic expressions, simplifying calculations and revealing hidden relationships. One of the most crucial properties is the power rule of logarithms, which states that logb(xp) = p * logb(x). This rule is particularly useful when dealing with exponents within logarithms, as it allows us to bring the exponent down as a coefficient, effectively simplifying the expression. In our case, we have log₃(271/4) from our earlier simplification. Applying the power rule, we can rewrite this as (1/4) * log₃(27).
The application of logarithmic properties in solving log₃√[4]{27} continues with the simplification of log₃(27). We recognize that 27 can be expressed as 3³, which means we are dealing with log₃(3³). Now, we can apply another key logarithmic property: the logarithm of a power with the same base. This property states that logb(bp) = p. In simpler terms, if the base of the logarithm is the same as the base of the exponential term within the logarithm, the expression simplifies to the exponent. Applying this to our expression, log₃(3³), we directly obtain 3. This simplification is a testament to the power of logarithms in unraveling exponential relationships.
Substituting the simplified value back into our expression, we have (1/4) * log₃(27) = (1/4) * 3. This final step involves a simple multiplication: (1/4) * 3 = 3/4. Therefore, the value of log₃√[4]{27} is 3/4. This confirms our previous solution and further demonstrates how logarithmic properties can streamline the problem-solving process. By strategically applying these properties, we can transform complex logarithmic expressions into more manageable forms, ultimately leading to a solution. The use of logarithmic properties not only simplifies calculations but also provides a deeper understanding of the structure and behavior of logarithms. These properties are indispensable tools in mathematics, enabling us to solve a wide range of problems involving exponential and logarithmic relationships with greater ease and efficiency. The mastery of these properties is crucial for anyone seeking to excel in mathematics and related fields, as they provide a foundation for advanced mathematical concepts and applications.
The Solution: 3/4
In summary, to find the value of log₃√[4]{27}, we systematically applied the definitions and properties of logarithms and exponents. We began by rewriting the radical expression √[4]{27} as 271/4, utilizing the relationship between radicals and fractional exponents. This transformation allowed us to work with exponents, which are often more manageable than radicals. Next, we recognized that 27 could be expressed as 3³, which is crucial because it relates the argument of the logarithm to its base. Substituting 3³ for 27, we obtained (3³)1/4. Applying the power of a power rule, we multiplied the exponents, resulting in 33/4. This simplification brought us closer to the solution by expressing the argument of the logarithm in terms of the base.
With the argument simplified to 33/4, we evaluated log₃(33/4). At this point, we invoked the fundamental definition of logarithms, which states that logb(bx) = x. In our case, the base is 3, and the argument is 3 raised to the power of 3/4. Therefore, the logarithm is simply the exponent, 3/4. Alternatively, we could have used the power rule of logarithms, which allows us to bring the exponent down as a coefficient: log₃(33/4) = (3/4) * log₃(3). Since log₃(3) equals 1, the expression simplifies to (3/4) * 1 = 3/4. This alternative approach reinforces the power and versatility of logarithmic properties in simplifying expressions.
Therefore, the value of log₃√[4]{27} is 3/4. This result underscores the interconnectedness of exponents and logarithms, as well as the importance of mastering their properties. By understanding these concepts and applying them strategically, we can tackle complex mathematical problems with confidence. The solution not only provides a numerical answer but also reinforces our grasp of the underlying mathematical principles, making it easier to apply these techniques to other problems. The journey of solving this logarithmic expression highlights the beauty and elegance of mathematics, where seemingly complex problems can be broken down into simpler steps through the application of fundamental rules and concepts. This approach is not only applicable to mathematics but also to problem-solving in various fields, where a systematic and logical approach is key to finding solutions.
Conclusion
In conclusion, understanding the value of expressions like log₃√[4]{27} is not just about arriving at a numerical answer; it’s about developing a deeper appreciation for the interplay between logarithms and exponents. Through a step-by-step approach, we successfully simplified the expression by leveraging the definitions and properties of logarithms and exponents. This process involved rewriting the radical as a fractional exponent, recognizing the relationship between the base of the logarithm and the argument, and strategically applying logarithmic properties like the power rule. The final result, 3/4, is a testament to the power of these mathematical tools in simplifying complex expressions.
The significance of mastering logarithms extends far beyond the confines of the mathematics classroom. Logarithms are fundamental to various scientific and engineering disciplines, serving as essential tools for modeling and analyzing phenomena that exhibit exponential behavior. From measuring sound intensity on the decibel scale to determining the acidity of a solution on the pH scale, logarithms provide a framework for understanding and quantifying the world around us. In computer science, logarithms are crucial for analyzing the efficiency of algorithms, particularly in sorting and searching. In finance, they are used to calculate compound interest and model investment growth. The ubiquity of logarithms in these diverse fields underscores their importance in both theoretical and practical applications.
The ability to solve logarithmic expressions is more than just a mathematical skill; it is a testament to one's problem-solving abilities and logical thinking. The process of breaking down a complex problem into manageable steps, identifying the relevant properties and definitions, and applying them strategically is a skill that transcends mathematics. It is a skill that is valuable in any field that requires critical thinking and problem-solving. Therefore, mastering logarithms is not just about learning a mathematical concept; it is about developing a powerful toolkit for tackling challenges in any domain. The journey of understanding logarithms is a journey of intellectual growth, fostering analytical thinking and problem-solving skills that are essential for success in an increasingly complex world.
