Logarithm Properties: Simplify $\log _6 \frac{2}{7 Z}$

When you're working with logarithms, understanding their properties is key to simplifying complex expressions. One of the most fundamental properties is the quotient rule for logarithms. This rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. In mathematical terms, this is expressed as: $\log _b \left( \frac{x}{y} \right) = \log _b (x) - \log _b (y)$. This property is incredibly useful because it allows us to break down a single logarithmic expression involving division into two separate logarithmic expressions. Imagine you have a complex fraction inside a logarithm; applying the quotient rule is often the first step to unraveling it and making it more manageable. We're assuming here that our base $b$ is positive and not equal to 1, and that both $x$ and $y$ are positive numbers, which is standard for logarithm definitions. This rule stems directly from the properties of exponents, specifically that $\frac{a^m}{a^n} = a^{m-n}$. If we consider $\log _b(x) = m$ and $\log _b(y) = n$, then $x = b^m$ and $y = b^n$. Therefore, $\frac{x}{y} = \frac{b^m}{b^n} = b^{m-n}$. Taking the logarithm base $b$ of both sides, we get $\log _b \left( \frac{x}{y} \right) = \log _b(b^{m-n}) = m-n$. Substituting back $m$ and $n$, we arrive at $\log _b \left( \frac{x}{y} \right) = \log _b(x) - \log _b(y)$. This foundational understanding empowers us to tackle expressions like the one we're about to explore.

Now, let's apply this powerful quotient rule for logarithms to our specific problem: $\log _6 \frac{2}{7 z}$. Here, our base $b$ is 6, the numerator $x$ is 2, and the denominator $y$ is $7z$. Since the problem statement specifies that variables represent positive numbers, we know that $7z$ is positive, and thus our conditions for applying the quotient rule are met. According to the rule, we can rewrite this expression as the difference of two logarithms: $\log _6 (2) - \log _6 (7z)$. We've successfully broken down the original expression into two simpler parts. Each of these new terms is a logarithm of a product or a number. It's important to note that while we've simplified the division aspect, the second term, $\log _6 (7z)$, still involves a product within the logarithm. This indicates that there might be further simplification possible using other logarithm properties, which we will explore next. The goal is always to expand or condense logarithmic expressions to a form that is easiest to work with, whether that means evaluating it or using it in further algebraic manipulations. Recognizing when and how to apply these rules is a skill that improves with practice, and this example provides a clear illustration of the initial step in such a process.

Continuing our journey with logarithm properties, we've arrived at $\log _6 (2) - \log _6 (7z)$. Notice the second term, $\log _6 (7z)$. This term is the logarithm of a product. Fortunately, there's another fundamental property of logarithms that helps us here: the product rule for logarithms. This rule states that the logarithm of a product is equal to the sum of the logarithms of the factors. Mathematically, it's written as: $\log _b (x imes y) = \log _b (x) + \log _b (y)$. Just like the quotient rule, this rule allows us to break down a single logarithm into simpler components. It’s derived from the exponent rule $a^m imes a^n = a^{m+n}$. If we let $\log _b(x) = m$ and $\log _b(y) = n$, then $x = b^m$ and $y = b^n$. Multiplying these gives us $x imes y = b^m imes b^n = b^{m+n}$. Taking the logarithm base $b$ of both sides yields $\log _b (x imes y) = \log _b (b^{m+n}) = m+n$. Substituting back $m$ and $n$, we get $\log _b (x imes y) = \log _b (x) + \log _b (y)$. This property is crucial for expanding expressions where numbers or variables are multiplied together inside a logarithm.

Applying the product rule for logarithms to our term $\log _6 (7z)$, we identify the base $b$ as 6, the first factor $x$ as 7, and the second factor $y$ as $z$. Both 7 and $z$ are positive numbers, satisfying the conditions for this rule. Therefore, we can rewrite $\log _6 (7z)$ as the sum of two logarithms: $\log _6 (7) + \log _6 (z)$. Now, we substitute this back into our expression from the previous step, which was $\log _6 (2) - \log _6 (7z)$. Replacing $\log _6 (7z)$ with its expanded form, we get: $\log _6 (2) - (\log _6 (7) + \log _6 (z))$. It is absolutely critical to use the parentheses here. The minus sign applies to the entire sum that replaces $\log _6 (7z)$. If we didn't use parentheses, we would incorrectly distribute the negative sign. Distributing the negative sign into the parentheses gives us the final expression: $\log _6 (2) - \log _6 (7) - \log _6 (z)$. This is the expression written as a difference of logarithms, as required. Each term is now a logarithm of a single number or a single variable, making the expression as expanded as it can be using these basic properties. The process involved identifying the structure of the original expression (a quotient) and then applying the appropriate rule (quotient rule). Within the resulting expression, we identified another structure (a product) and applied another rule (product rule), carefully handling the distribution of the negative sign. This systematic approach is the hallmark of working effectively with logarithmic identities.

In summary, to write the expression $\log _6 \frac{2}{7 z}$ as a sum or difference of logarithms, we systematically applied two fundamental logarithm properties. First, we used the quotient rule, $\log _b \left( \frac{x}{y} \right) = \log _b (x) - \log _b (y)$, to break down the fraction inside the logarithm. This transformed $\log _6 \frac{2}{7 z}$ into $\log _6 (2) - \log _6 (7z)$. Second, we recognized that the term $\log _6 (7z)$ involved a product and applied the product rule, $\log _b (x imes y) = \log _b (x) + \log _b (y)$, to expand it into $\log _6 (7) + \log _6 (z)$. When substituting this back into our expression, it was crucial to maintain the correct sign. The original expression became $\log _6 (2) - (\log _6 (7) + \log _6 (z))$. Finally, by distributing the negative sign, we arrived at the desired form: $\mathbf{\log _6 (2) - \log _6 (7) - \log _6 (z)}$. This final expression is a difference of logarithms, where each logarithm contains only a single number or a single variable. This process demonstrates the power of logarithmic identities in simplifying and expanding expressions, making them more amenable to further analysis or calculation. Remember, the key is to identify the operations (division, multiplication) within the logarithm and match them with the corresponding rules. Always pay close attention to parentheses and the distribution of negative signs, as these are common pitfalls. Mastering these properties will serve you well in algebra and calculus. For further exploration of logarithm properties and their applications, you can visit reliable sources such as Khan Academy or Paul's Online Math Notes, which offer comprehensive explanations and practice problems.