The Key To Simplifying Logarithmic Expressions Unveiling The Property Behind $\log _b(b^{x+y})$

When diving into the fascinating realm of logarithms, understanding their fundamental properties is crucial. Among these, the product property of logarithms stands out as a powerful tool for simplifying complex expressions and solving logarithmic equations. To truly grasp this property, it's essential to understand the underlying principles that make it work. One key aspect of the proof involves simplifying the expression $\log _b(b^{x+y})$ to $x+y$. But what exactly is the justification for this step? Let's delve into the world of logarithms and uncover the property that allows this simplification.

Understanding Logarithms: The Foundation

Before we tackle the specific question at hand, it's crucial to have a solid understanding of what logarithms are and how they relate to exponential functions. In simple terms, a logarithm answers the question: "To what power must we raise the base to get a certain number?" Mathematically, this is expressed as:

$\log _b(a) = c$ if and only if $b^c = a$

Here:

• $b$ is the base of the logarithm.
• $a$ is the argument of the logarithm (the number we want to find the logarithm of).
• $c$ is the exponent to which we must raise the base $b$ to obtain $a$.

For example, $\log _{10}(100) = 2$ because $10^2 = 100$. Similarly, $\log _2(8) = 3$ because $2^3 = 8$. This fundamental relationship between logarithms and exponents is the bedrock upon which all logarithmic properties are built.

The Product Property of Logarithms

The product property of logarithms is one of the core rules governing how logarithms behave. It states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. In mathematical notation:

$\log _b(mn) = \log _b(m) + \log _b(n)$

Where:

• $b$ is the base of the logarithm.
• $m$ and $n$ are positive real numbers.

This property is incredibly useful for breaking down complex logarithmic expressions into simpler terms. For instance, we can use the product property to rewrite $\log _2(16 imes 32)$ as $\log _2(16) + \log _2(32)$, which is much easier to evaluate. But how do we prove this property? That's where the initial question about simplifying $\log _b(b^{x+y})$ comes into play.

Proving the Product Property: A Step-by-Step Approach

To prove the product property, we typically start by expressing the arguments of the logarithms as exponential terms. Let's assume:

$m = b^x$ and $n = b^y$

This implies that:

$\log _b(m) = x$ and $\log _b(n) = y$

Now, let's consider the product $mn$:

$mn = b^x imes b^y$

Using the rule of exponents that states $b^x imes b^y = b^{x+y}$, we can rewrite the product as:

$mn = b^{x+y}$

Now, let's take the logarithm base $b$ of both sides:

$\log _b(mn) = \log _b(b^{x+y})$

This is where the critical step of simplifying the expression $\log _b(b^{x+y})$ comes into play. The question asks us to identify the property that justifies this simplification.

The Key Property: $\log _b(b^c) = c$

The property that allows us to simplify $\log _b(b^{x+y})$ to $x+y$ is the fundamental logarithmic identity: $\log _b(b^c) = c$. This identity is a direct consequence of the definition of a logarithm. It essentially states that the logarithm of a base raised to an exponent is simply the exponent itself.

In our case, we have $\log _b(b^{x+y})$. Applying the property $\log _b(b^c) = c$, where $c = x+y$, we get:

$\log _b(b^{x+y}) = x+y$

This simplification is the linchpin in the proof of the product property. Now, let's continue with the proof.

Completing the Proof

Substituting the simplified expression back into our equation, we have:

$\log _b(mn) = x+y$

Recall that we defined $x = \log _b(m)$ and $y = \log _b(n)$. Substituting these back into the equation, we get:

$\log _b(mn) = \log _b(m) + \log _b(n)$

This is precisely the product property of logarithms! We have successfully proven the property by leveraging the fundamental logarithmic identity $\log _b(b^c) = c$ to simplify the expression $\log _b(b^{x+y})$.

Why is This Property So Important?

The product property of logarithms is not just a theoretical curiosity; it has practical applications in various fields, including:

• Simplifying Complex Expressions: Logarithms can transform complex multiplication problems into simpler addition problems. This is particularly useful when dealing with very large or very small numbers.
• Solving Logarithmic Equations: The product property allows us to combine logarithmic terms, which is often a crucial step in solving logarithmic equations.
• Calculus and Engineering: Logarithmic functions and their properties are fundamental in calculus and engineering applications, such as analyzing exponential growth and decay.
• Computer Science: Logarithms are used extensively in computer science for analyzing algorithms and data structures.

Examining the Answer Choices

Now that we have a thorough understanding of the proof and the relevant property, let's revisit the original question and the answer choices:

The proof for the product property of logarithms requires simplifying the expression $\log _b(b^{x+y})$ to $x+y$. Which property is used to justify this step?

• A. $b^x \cdot b^y=b^{x+y}$
• B. Substitution
• C. $\log _b(b^c)=c$
• D. Discussion category: mathematics

Let's analyze each option:

• A. $b^x \cdot b^y=b^{x+y}$: This is the rule of exponents that we used earlier in the proof to combine $b^x$ and $b^y$. While important for the overall proof, it doesn't directly justify the simplification of $\log _b(b^{x+y})$.
• B. Substitution: Substitution was used throughout the proof, but it's a general technique, not a specific property that justifies the simplification.
• C. $\log _b(b^c)=c$: This is the fundamental logarithmic identity that we identified as the key to simplifying $\log _b(b^{x+y})$. It directly states that the logarithm of a base raised to an exponent is the exponent itself.
• D. Discussion category: mathematics: This is not a mathematical property.

Therefore, the correct answer is C. $\log _b(b^c)=c$.

Conclusion

The proof of the product property of logarithms hinges on the ability to simplify the expression $\log _b(b^{x+y})$ to $x+y$. This simplification is justified by the fundamental logarithmic identity $\log _b(b^c) = c$. Understanding this property and its role in the proof not only deepens our understanding of logarithms but also equips us with a powerful tool for simplifying expressions and solving logarithmic equations. The product property, along with other logarithmic properties, is a cornerstone of mathematics and finds applications in various scientific and engineering disciplines. By mastering these concepts, you unlock a deeper appreciation for the elegance and utility of logarithms.