Rewrite Log(21x) Using The Product Rule A Step-by-Step Guide
In the realm of mathematics, logarithms serve as a powerful tool for simplifying complex calculations and unveiling hidden relationships between numbers. The product rule for logarithms is one such rule that allows us to rewrite the logarithm of a product as the sum of logarithms, providing a valuable technique for manipulating logarithmic expressions. In this article, we'll delve into the product rule for logarithms, demonstrate its application by rewriting the expression log(21x) as a sum of logarithms, and explore the broader significance of this rule in various mathematical contexts.
Understanding the Product Rule for Logarithms
The product rule for logarithms is a fundamental property that states that the logarithm of the product of two or more numbers is equal to the sum of the logarithms of those numbers. Mathematically, this rule can be expressed as follows:
logb(xy) = logb(x) + logb(y)
where:
- b is the base of the logarithm (b > 0, b ≠1)
- x and y are positive real numbers
This rule holds true for any base b, as long as it adheres to the conditions mentioned above. The product rule stems from the very definition of logarithms and their intimate connection with exponents. Recall that logarithms are essentially the inverse operation of exponentiation. The product rule for logarithms mirrors the rule for exponents that states that when multiplying exponential terms with the same base, we add the exponents.
To illustrate this connection, let's consider the following:
Let x = bm and y = bn
Then, logb(x) = m and logb(y) = n
Now, consider the product xy:
xy = bm * bn = b(m+n)
Taking the logarithm of both sides with base b, we get:
logb(xy) = logb(b(m+n))
Using the property that logb(ba) = a, we have:
logb(xy) = m + n
Substituting back the values of m and n, we get:
logb(xy) = logb(x) + logb(y)
This derivation clearly demonstrates the link between the product rule for logarithms and the underlying principles of exponents.
The product rule for logarithms provides a valuable tool for simplifying logarithmic expressions. By breaking down the logarithm of a product into the sum of individual logarithms, we can often manipulate expressions more easily, solve equations, and gain a deeper understanding of the relationships between variables. The rule is particularly useful when dealing with complex expressions involving multiple factors within a logarithm.
Applying the Product Rule to Log(21x)
Now, let's apply the product rule for logarithms to the expression log(21x). We aim to rewrite this expression as a sum of logarithms, effectively breaking down the product within the logarithm. To do this, we first recognize that 21x is the product of 21 and x. Thus, according to the product rule, we can rewrite log(21x) as the sum of log(21) and log(x):
log(21x) = log(21) + log(x)
This is a direct application of the product rule, where we have separated the logarithm of the product into the sum of the logarithms of the individual factors. However, we can further simplify this expression by recognizing that 21 itself is a product of two prime numbers, 3 and 7. This allows us to apply the product rule once more to log(21):
log(21) = log(3 * 7)
Applying the product rule again, we get:
log(3 * 7) = log(3) + log(7)
Now, we can substitute this back into our original expression:
log(21x) = log(21) + log(x) = log(3) + log(7) + log(x)
Therefore, we have successfully rewritten log(21x) as the sum of logarithms: log(3) + log(7) + log(x). This expression now represents the original logarithm as a sum of logarithms of its prime factors and the variable x. This form can be advantageous in various mathematical contexts, such as simplifying equations, analyzing relationships between variables, and performing numerical calculations.
Do Not Evaluate the Logarithm
It is crucial to note that the instructions explicitly state not to evaluate the logarithm. This means we should not attempt to find the numerical values of log(3), log(7), or log(x). Our goal is solely to rewrite the expression using the product rule, and we have achieved that by expressing log(21x) as log(3) + log(7) + log(x). Evaluating these logarithms would involve finding the powers to which the base (which is 10 for the common logarithm if no base is specified) must be raised to obtain 3, 7, and x, respectively. While these values can be approximated using calculators or logarithmic tables, doing so would deviate from the instructions and negate the purpose of the exercise, which is to demonstrate the application of the product rule itself.
By leaving the logarithms unevaluated, we retain the symbolic representation of the logarithmic relationship, highlighting the structure of the expression and the way it has been transformed using the product rule. This is often more valuable in mathematical manipulations and problem-solving than obtaining a numerical approximation.
Comparison with Log7(3) + 1 + Log7(2)
The original problem statement also includes the expression log7(3) + 1 + log7(2). While this expression is not directly related to rewriting log(21x) using the product rule, it provides an opportunity to illustrate other logarithmic properties and simplifications. Let's examine this expression further.
We have log7(3) + 1 + log7(2). To simplify this expression, we can use the property that logb(b) = 1. In this case, since the base of the logarithms is 7, we can rewrite 1 as log7(7):
log7(3) + 1 + log7(2) = log7(3) + log7(7) + log7(2)
Now, we can apply the product rule in reverse. The product rule states that logb(x) + logb(y) = logb(xy). We can extend this rule to multiple terms:
logb(x) + logb(y) + logb(z) = logb(xyz)
Applying this to our expression, we get:
log7(3) + log7(7) + log7(2) = log7(3 * 7 * 2)
Simplifying the product inside the logarithm:
log7(3 * 7 * 2) = log7(42)
Therefore, the expression log7(3) + 1 + log7(2) can be simplified to log7(42). This demonstrates how the product rule, along with other logarithmic properties, can be used to combine and simplify logarithmic expressions. The key difference between this simplification and rewriting log(21x) is that here, we are combining logarithms into a single logarithm, while in the previous case, we were expanding a single logarithm into a sum of logarithms.
Significance of the Product Rule for Logarithms
The product rule for logarithms is not merely a mathematical curiosity; it plays a crucial role in various fields of science, engineering, and finance. Its significance stems from its ability to transform multiplicative relationships into additive ones, often simplifying complex calculations and revealing underlying patterns.
In scientific and engineering applications, the product rule is frequently used in areas such as signal processing, acoustics, and seismology. In these fields, signals and waves are often represented as products of different components. By applying the product rule for logarithms, engineers and scientists can analyze these signals by examining the individual logarithmic components, making it easier to identify patterns, filter noise, and extract relevant information.
For instance, in acoustics, the intensity of sound is often measured on a logarithmic scale using decibels. The product rule allows us to analyze the combined intensity of multiple sound sources by simply adding their individual decibel levels. Similarly, in seismology, the magnitude of earthquakes is measured using the Richter scale, which is a logarithmic scale. The product rule helps seismologists to understand the combined energy released by multiple seismic events.
In finance, the product rule finds applications in areas such as compound interest calculations and portfolio management. Compound interest involves the repeated multiplication of the principal amount by a growth factor. By taking the logarithm of the compound interest formula, we can transform the multiplicative relationship into an additive one, making it easier to calculate the future value of an investment or the time required for an investment to reach a certain target.
In portfolio management, the return on a portfolio is often expressed as a product of the returns on individual assets. The product rule allows investors to analyze the overall portfolio return by examining the individual logarithmic returns, providing insights into the contribution of each asset to the overall portfolio performance. This can help investors to make informed decisions about asset allocation and risk management.
Beyond these specific applications, the product rule for logarithms is a fundamental tool in mathematical analysis and problem-solving. It provides a powerful technique for manipulating logarithmic expressions, solving equations, and understanding the relationships between variables. Its versatility and wide-ranging applicability make it an indispensable part of the mathematical toolkit for students, researchers, and professionals alike.
Conclusion
The product rule for logarithms is a cornerstone of logarithmic operations, enabling us to rewrite the logarithm of a product as the sum of logarithms. By applying this rule, we can simplify complex expressions, solve equations, and gain deeper insights into the relationships between variables. In this article, we successfully rewrote log(21x) as log(3) + log(7) + log(x) using the product rule, demonstrating the practical application of this important logarithmic property. Furthermore, we explored the broader significance of the product rule in various fields, highlighting its versatility and importance in mathematics and beyond. Understanding and mastering the product rule is essential for anyone seeking to delve deeper into the world of logarithms and their myriad applications.
