Express Log₅(14√x/y) As Sums And Differences Of Logarithms

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In the realm of mathematics, logarithms serve as powerful tools for simplifying complex calculations and revealing hidden relationships between numbers. They are the inverse operation to exponentiation, allowing us to express numbers as powers of a specific base. When dealing with logarithmic expressions, we often encounter scenarios where we need to manipulate them to gain deeper insights or simplify further calculations. One such manipulation involves expressing a single logarithm as a sum or difference of multiple logarithms, leveraging key properties of logarithms. In this comprehensive exploration, we will delve into the process of rewriting the logarithmic expression log₅(14√x/y) as a combination of sums and differences of logarithms, while also expressing any powers as factors. This process will not only enhance our understanding of logarithmic properties but also equip us with the ability to tackle more intricate logarithmic problems.

Understanding Logarithmic Properties

Before we embark on the transformation of the given expression, it is crucial to solidify our understanding of the fundamental properties of logarithms that govern these manipulations. These properties act as the building blocks for rewriting logarithmic expressions and are essential for ensuring the accuracy of our transformations. Let's delve into the core properties that will guide our journey:

  1. Product Rule: The logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this property is expressed as: logb(mn) = logb(m) + logb(n). This rule allows us to break down complex expressions involving multiplication within a logarithm into simpler sums of logarithms.

  2. Quotient Rule: The logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator. Expressed mathematically, this property states: logb(m/n) = logb(m) - logb(n). This rule enables us to separate expressions involving division within a logarithm into differences of logarithms.

  3. Power Rule: The logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number. Mathematically, this property is represented as: logb(mp) = p * logb(m). This rule is particularly useful when dealing with exponents within logarithms, allowing us to bring the exponent down as a coefficient.

  4. Logarithm of a Root: The logarithm of the nth root of a number can be expressed as the logarithm of the number raised to the power of 1/n. Mathematically, this can be represented as: logb(√[n]m) = logb(m^(1/n)) = (1/n) * logb(m). This property provides a direct link between logarithms and roots, enabling us to handle expressions involving roots within logarithms.

With these properties firmly in place, we are now well-equipped to tackle the task of rewriting the logarithmic expression log₅(14√x/y) as a sum and/or difference of logarithms, while also expressing powers as factors. The journey ahead promises to be an insightful exploration of the power and versatility of logarithmic properties.

Applying Logarithmic Properties to Expand log₅(14√x/y)

Now, let's embark on the journey of rewriting the given logarithmic expression, log₅(14√x/y), as a sum and/or difference of logarithms while expressing powers as factors. This process involves carefully applying the logarithmic properties we've discussed to systematically break down the expression into simpler components.

  1. Step 1: Applying the Quotient Rule

Our initial focus will be on the overall structure of the expression, which involves a quotient. The quotient rule of logarithms, which states logb(m/n) = logb(m) - logb(n), provides the perfect tool to separate the numerator and denominator. Applying this rule to our expression, we get:

log₅(14√x/y) = log₅(14√x) - log₅(y)

This step effectively separates the expression into two distinct logarithmic terms, each representing a part of the original quotient. We've successfully removed the division operation from within the logarithm.

  1. Step 2: Applying the Product Rule

Next, we turn our attention to the first term, log₅(14√x), which involves a product. The product rule of logarithms, which states logb(mn) = logb(m) + logb(n), comes to our aid in separating the factors within the product. Applying this rule, we obtain:

log₅(14√x) = log₅(14) + log₅(√x)

This step further breaks down the expression, isolating the individual factors within the product. We now have three logarithmic terms, each representing a distinct component of the original expression.

  1. Step 3: Expressing the Square Root as a Power

We notice that the second term, log₅(√x), involves a square root. To apply the power rule effectively, we need to express the square root as a power. Recall that the square root of a number is equivalent to raising that number to the power of 1/2. Therefore, we can rewrite the term as:

log₅(√x) = log₅(x^(1/2))

This transformation sets the stage for applying the power rule in the next step.

  1. Step 4: Applying the Power Rule

Now, we can apply the power rule of logarithms, which states logb(mp) = p * logb(m), to the term log₅(x^(1/2)). This rule allows us to bring the exponent (1/2 in this case) down as a coefficient:

log₅(x^(1/2)) = (1/2) * log₅(x)

This step completes the transformation of the square root term, expressing it as a product of a constant and a logarithm.

  1. Step 5: Combining the Results

Finally, we combine all the individual transformations to obtain the fully expanded expression:

log₅(14√x/y) = log₅(14) + log₅(√x) - log₅(y)
= log₅(14) + log₅(x^(1/2)) - log₅(y)
= log₅(14) + (1/2) * log₅(x) - log₅(y)

Thus, we have successfully rewritten the original logarithmic expression as a sum and difference of logarithms, while also expressing the power as a factor. The final expanded form is:

log₅(14√x/y) = log₅(14) + (1/2)log₅(x) - log₅(y)

This expanded form provides a clearer representation of the expression's components and can be particularly useful in various mathematical contexts, such as solving equations or simplifying complex expressions.

Detailed Breakdown of the Expanded Form

The expanded form of the logarithmic expression, log₅(14√x/y) = log₅(14) + (1/2)log₅(x) - log₅(y), offers a valuable perspective on the individual components that contribute to the overall value of the expression. Let's delve into a detailed breakdown of each term and its significance:

  1. log₅(14): This term represents the logarithm of 14 with base 5. In essence, it answers the question,