Expressing 3 Log2 X - (log2 3 - Log2(x+4)) As A Single Logarithm

In the realm of mathematics, particularly when dealing with logarithms, the ability to manipulate and simplify expressions is paramount. Logarithmic expressions, while seemingly complex at first glance, adhere to a set of rules and properties that allow us to rewrite them in more concise and manageable forms. This article delves into the process of condensing a given logarithmic expression into a single logarithm, a skill that is not only fundamental in mathematical problem-solving but also crucial in various scientific and engineering applications.

The given expression, $3 \log _2 x-\left(\log _2 3-\log _2(x+4)\right)$, presents a combination of logarithmic terms that need to be simplified using logarithmic properties. The primary goal is to express this expression as a single logarithm, which means combining all the individual logarithmic terms into one. This involves utilizing properties such as the power rule, the quotient rule, and the product rule of logarithms. Each of these rules plays a specific role in the simplification process, allowing us to manipulate the expression step by step until we arrive at the desired single logarithmic form.

Understanding the properties of logarithms is key to tackling such problems. The power rule, for instance, allows us to move coefficients within the logarithm as exponents. The quotient rule enables us to combine logarithms that are subtracted, while the product rule helps in combining logarithms that are added. By applying these rules judiciously, we can transform the initial expression into a more streamlined and elegant form, ultimately expressing it as a single logarithmic term. This process not only simplifies the expression but also enhances our understanding of the underlying logarithmic relationships.

Breaking Down the Expression

To effectively simplify the expression $3 \log _2 x-\left(\log _2 3-\log _2(x+4)\right)$, we need to break it down step by step, applying the relevant logarithmic properties at each stage. This methodical approach ensures that we don't miss any crucial steps and that the simplification is carried out correctly. Let's begin by focusing on the initial structure of the expression and identifying the operations that need to be performed.

Applying the Power Rule

The first term in the expression is $3 \log _2 x$. According to the power rule of logarithms, which states that $a \log_b c = \log_b (c^a)$, we can rewrite this term by moving the coefficient 3 as an exponent of x. This transformation yields $\log _2 (x^3)$. The power rule is a fundamental tool in simplifying logarithmic expressions, as it allows us to deal with coefficients that multiply logarithms by incorporating them into the argument of the logarithm itself. This step is crucial as it sets the stage for further simplification using other logarithmic properties.

By applying the power rule, we have effectively reduced the complexity of the first term, making it easier to combine with other logarithmic terms later on. This rule is particularly useful when dealing with expressions that involve multiple logarithmic terms with coefficients, as it allows us to consolidate these coefficients into the arguments of the logarithms. The result, $\log _2 (x^3)$, is a more compact and manageable form of the original term, which is a key step towards expressing the entire expression as a single logarithm.

Dealing with Parentheses and Negative Signs

Next, we need to address the expression within the parentheses: $\left(\log _2 3-\log _2(x+4)\right)$. The negative sign preceding the second term, $\log _2(x+4)$, indicates that we are dealing with a subtraction of logarithms. To simplify this, we can apply the quotient rule of logarithms, which states that $\log_b a - \log_b c = \log_b \left(\frac{a}{c}\right)$. This rule allows us to combine two logarithms that are being subtracted into a single logarithm of a quotient.

Applying the quotient rule to the terms within the parentheses, we get $\log _2 3 - \log _2(x+4) = \log _2 \left(\frac{3}{x+4}\right)$. This transformation is a significant step in simplifying the overall expression, as it reduces two logarithmic terms into one. The quotient rule is a powerful tool for condensing logarithmic expressions, especially when dealing with differences between logarithms. By using this rule, we can effectively combine multiple logarithmic terms into a single term, making the expression easier to manipulate and solve.

Now, we need to consider the negative sign outside the parentheses. The original expression was $3 \log _2 x-\left(\log _2 3-\log _2(x+4)\right)$, which we have now simplified to $3 \log _2 x - \log _2 \left(\frac{3}{x+4}\right)$. Distributing the negative sign, we get $\log _2 (x^3) - \log _2 \left(\frac{3}{x+4}\right)$. This step is crucial as it prepares us to combine the remaining logarithmic terms into a single logarithm.

Combining Logarithmic Terms

With the expression now in the form $\log _2 (x^3) - \log _2 \left(\frac{3}{x+4}\right)$, we are ready to apply the quotient rule again to combine these two logarithms into a single term. This step is essential for achieving the ultimate goal of expressing the entire original expression as a single logarithm. The quotient rule, as we've seen, is a powerful tool for simplifying differences between logarithms, and its application here will consolidate the expression into a more manageable form.

Applying the Quotient Rule Again

As a reminder, the quotient rule of logarithms states that $\log_b a - \log_b c = \log_b \left(\frac{a}{c}\right)$. In our case, $a = x^3$ and $c = \frac{3}{x+4}$. Applying the quotient rule, we get:

$\log _2 (x^3) - \log _2 \left(\frac{3}{x+4}\right) = \log _2 \left(\frac{x^3}{\frac{3}{x+4}}\right)$

This step is a direct application of the quotient rule, where we have combined the two logarithms into a single logarithm of a quotient. The resulting expression, $\log _2 \left(\frac{x^3}{\frac{3}{x+4}}\right)$, is closer to the desired form of a single logarithm, but it still requires further simplification to remove the complex fraction within the logarithm's argument.

Simplifying the Fraction

To simplify the complex fraction inside the logarithm, we need to divide $x^3$ by $\frac{3}{x+4}$. Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we can rewrite the expression as:

$\log _2 \left(\frac{x^3}{\frac{3}{x+4}}\right) = \log _2 \left(x^3 \cdot \frac{x+4}{3}\right)$

This step involves a basic algebraic manipulation, where we have converted the division by a fraction into multiplication by its reciprocal. This transformation is crucial for simplifying the expression and making it easier to combine the terms within the logarithm. The result, $\log _2 \left(x^3 \cdot \frac{x+4}{3}\right)$, is a more simplified form of the complex fraction, but we still need to combine the terms within the parentheses to achieve the final single logarithmic form.

Final Simplification

Now, we can simplify the expression inside the logarithm by multiplying the terms: $x^3 \cdot \frac{x+4}{3} = \frac{x^3(x+4)}{3}$. Substituting this back into the logarithm, we get:

$\log _2 \left(x^3 \cdot \frac{x+4}{3}\right) = \log _2 \left(\frac{x^3(x+4)}{3}\right)$

This final simplification step involves a straightforward multiplication of terms, resulting in a single fraction within the logarithm's argument. The expression $\log _2 \left(\frac{x^3(x+4)}{3}\right)$ represents the original expression written as a single logarithm, which is the ultimate goal of our simplification process.

Conclusion

By systematically applying the properties of logarithms, we have successfully transformed the original expression, $3 \log _2 x-\left(\log _2 3-\log _2(x+4)\right)$, into a single logarithm: $\log _2 \left(\frac{x^3(x+4)}{3}\right)$. This process involved the strategic use of the power rule, the quotient rule, and basic algebraic manipulations. Understanding and applying these logarithmic properties is essential for simplifying complex expressions and solving mathematical problems involving logarithms.

The correct answer is A. $\log _2\left[\frac{x^3(x+4)}{3}\right]$. This result showcases the power of logarithmic properties in simplifying expressions and highlights the importance of a step-by-step approach in mathematical problem-solving. The ability to manipulate logarithmic expressions is a valuable skill in various fields, including mathematics, science, and engineering, making this type of simplification a fundamental concept to master.