Simplify Logarithmic Expression: Sum/Difference Of Logs

Hey guys! Let's dive into the world of logarithms and tackle a common problem: expressing a complex logarithmic expression as a sum or difference of simpler logarithms, all while eliminating those pesky exponents. Today, we're going to break down the expression (\log \sqrt[7]{\frac{y^{14} w^{15}}{x{15}}}) step-by-step. Buckle up, because by the end of this article, you'll be a log-simplifying pro!

Understanding the Initial Expression

Before we jump into the nitty-gritty, let's make sure we understand what we're dealing with. Our mission is to simplify the expression:

${ \log \sqrt[7]{\frac{y^{14} w^{15}}{x^{15}}} }$

This might look a little intimidating at first glance, but don't worry! We're going to use some key properties of logarithms to make it much easier to handle. We'll focus on transforming this complex expression into a sum and difference of individual logarithmic terms without any exponents inside the log. This involves applying several logarithmic properties, which we will discuss in detail. Let's get started!

Breaking Down the Components

The expression involves a logarithm of a seventh root of a fraction. The numerator of the fraction contains ${y^{14}}$ and ${w^{15}}$, while the denominator contains ${x^{15}}$. Our main tools will be the power rule, the quotient rule, and the product rule of logarithms. These rules allow us to manipulate logarithmic expressions by dealing with exponents, division, and multiplication, respectively. To kick things off, we'll first deal with the seventh root. Remember, roots can be expressed as fractional exponents, which is the first trick up our sleeve to simplify this expression. Understanding each component is crucial for simplifying complex logarithmic expressions, so let’s break down the process step by step.

Step 1: Converting the Root to a Fractional Exponent

The first thing we need to do is tackle that seventh root. Remember that a root can be rewritten as a fractional exponent. In this case, the seventh root is equivalent to raising the entire fraction to the power of ${\frac{1}{7}}$. So, we can rewrite our expression as:

${ \log \left(\frac{y^{14} w^{15}}{x^{15}}\right)^{\frac{1}{7}} }$

This is a crucial step because it sets us up to use the power rule of logarithms, which is our next key move. By converting the root into a fractional exponent, we make the entire expression more manageable and ready for further simplification. This transformation allows us to bring the exponent outside the logarithm, a significant step towards achieving our goal of expressing the logarithm as a sum or difference.

Power Rule of Logarithms

This step utilizes the power rule of logarithms, which states that ${\log_b(a^c) = c \log_b(a)}$. This rule is extremely helpful when simplifying logarithmic expressions that contain exponents, as it allows us to move the exponent from within the logarithm to a coefficient outside the logarithm. This is precisely what we aim to do here, as it simplifies the overall expression and makes it easier to work with. Let's see how this rule applies in our next step.

Step 2: Applying the Power Rule of Logarithms

Now that we've expressed the root as a fractional exponent, we can use the power rule of logarithms. The power rule states that ${\log_b(a^c) = c \log_b(a)}$. Applying this rule to our expression, we get:

${ \frac{1}{7} \log \left(\frac{y^{14} w^{15}}{x^{15}}\right) }$

See how we've moved the ${\frac{1}{7}}$ exponent outside the logarithm? This is a big win! Now, we're left with a simpler logarithmic expression inside the parentheses. The next step involves breaking down the fraction inside the logarithm using another important logarithmic property: the quotient rule. By applying the power rule, we've significantly reduced the complexity of the expression, paving the way for further simplification using other logarithmic rules.

Importance of the Power Rule

The power rule is a cornerstone of logarithmic simplification. It allows us to deal with exponents directly, transforming what might seem like a complex expression into something much more manageable. Without this rule, simplifying expressions like the one we're working on would be significantly more challenging. Understanding and applying the power rule effectively is crucial for anyone working with logarithms.

Step 3: Using the Quotient Rule of Logarithms

Next up, we'll use the quotient rule of logarithms to break down the fraction inside the logarithm. The quotient rule states that ${\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)}$. This rule allows us to separate the logarithm of a quotient into the difference of two logarithms. Applying this to our expression, we get:

${ \frac{1}{7} \left[ \log(y^{14} w^{15}) - \log(x^{15}) \right] }$

Notice how the fraction has been transformed into a difference of two logarithmic terms. This is another significant simplification. Now, we have two separate logarithms, but the first one still contains a product. To deal with this, we'll use the product rule of logarithms in the next step. By strategically applying the quotient rule, we've taken another step towards our goal of expressing the original logarithm as a sum or difference of simpler terms.

Expanding the Expression

Expanding the expression using the quotient rule helps in isolating the individual components, making it easier to apply further logarithmic properties. This step is crucial for untangling complex logarithmic expressions, allowing us to work with each part separately and simplify them more effectively.

Step 4: Applying the Product Rule of Logarithms

The first logarithmic term in our expression, ${\log(y^{14} w^{15})}$, involves a product. To simplify this, we'll use the product rule of logarithms. The product rule states that ${\log_b(mn) = \log_b(m) + \log_b(n)}$. Applying this rule, we can rewrite the term as:

${ \log(y^{14} w^{15}) = \log(y^{14}) + \log(w^{15}) }$

Now, substituting this back into our main expression, we get:

${ \frac{1}{7} \left[ \log(y^{14}) + \log(w^{15}) - \log(x^{15}) \right] }$

We've successfully broken down the product into a sum of logarithms. This is another key step in our simplification process. Next, we'll deal with the exponents inside the logarithms using, you guessed it, the power rule again! The strategic use of the product rule allows us to further dissect the original logarithmic expression into manageable parts.

Why the Product Rule Matters

The product rule is essential for dealing with expressions where logarithmic terms are multiplied together. By converting multiplication inside a logarithm into addition outside the logarithm, we can simplify complex expressions and make them easier to work with. This rule, alongside the quotient and power rules, forms the foundation of logarithmic simplification.

Step 5: Using the Power Rule Again

We're not done with the power rule yet! We still have exponents inside the logarithms in our expression. Let's apply the power rule ${\log_b(a^c) = c \log_b(a)}$ to each term:

${ \log(y^{14}) = 14 \log(y)\\ \log(w^{15}) = 15 \log(w)\\ \log(x^{15}) = 15 \log(x) }$

Substituting these back into our expression, we get:

${ \frac{1}{7} \left[ 14 \log(y) + 15 \log(w) - 15 \log(x) \right] }$

We've successfully moved all the exponents outside the logarithms. Now, all that's left is to distribute the ${\frac{1}{7}}$ and simplify. This repeated application of the power rule demonstrates its versatility and importance in logarithmic simplification.

Consistency is Key

Applying the power rule consistently is crucial for fully simplifying logarithmic expressions. By systematically addressing exponents, we can break down even the most complex logarithms into their simplest forms. This step prepares us for the final simplification, where we distribute the remaining coefficient.

Step 6: Distributing and Simplifying

Finally, let's distribute the ${\frac{1}{7}}$ across all the terms inside the brackets:

${ \frac{1}{7} \left[ 14 \log(y) + 15 \log(w) - 15 \log(x) \right] = \frac{14}{7} \log(y) + \frac{15}{7} \log(w) - \frac{15}{7} \log(x) }$

Now, simplify the fractions:

${ \frac{14}{7} \log(y) = 2 \log(y) }$

So, our final simplified expression is:

${ 2 \log(y) + \frac{15}{7} \log(w) - \frac{15}{7} \log(x) }$

We've done it! We've successfully expressed the original logarithmic expression as a sum and difference of logarithms with no exponents. Give yourself a pat on the back!

Final Thoughts on Simplification

The final distribution and simplification are the finishing touches that bring our hard work to fruition. By carefully distributing the coefficient and simplifying fractions, we arrive at the most concise form of the expression. This final step is a testament to the power of systematic simplification.

Conclusion

Simplifying logarithmic expressions might seem tricky at first, but by breaking it down step-by-step and using the key properties of logarithms – the power rule, the quotient rule, and the product rule – you can conquer any log problem! Remember, practice makes perfect. So, keep at it, and you'll become a log-simplification master in no time. The final expression, ${2 \log(y) + \frac{15}{7} \log(w) - \frac{15}{7} \log(x)}$, is the fully simplified form of our original expression. Great job, guys!

Mastering Logarithms

Mastering logarithms involves understanding and applying the fundamental rules consistently. The journey from a complex logarithmic expression to its simplified form requires patience, practice, and a strategic approach. By following these steps and understanding the underlying principles, you can confidently tackle any logarithmic simplification problem.