Express $3 \log _5 X+4 \log _5 Y$ As A Single Logarithm

Aug 1, 2025 by ADMIN 56 views

Expressing Multiple Logarithms as a Single Logarithm

Understanding Logarithmic Properties

Before diving into the problem, let's refresh our understanding of the logarithmic properties that we'll be using. These properties are essential for manipulating logarithmic expressions and combining them into a single logarithm. Logarithmic properties are the cornerstone of simplifying and solving logarithmic equations and expressions. Mastering these properties allows us to navigate complex mathematical problems with greater ease and precision. Remember, the goal is to condense multiple logarithmic terms into a single, simplified expression.

Power Rule

The power rule is our first key tool. It states that $\log_b(x^p) = p \log_b(x)$ . In simpler terms, if you have a logarithm of a number raised to a power, you can bring the power down and multiply it by the logarithm. This rule is incredibly useful for dealing with coefficients in front of logarithms. For instance, if we have an expression like $2\log_3(5)$ , we can rewrite it as $\log_3(5^2)$ , which simplifies to $\log_3(25)$ . The power rule essentially allows us to move exponents inside the logarithm, making it easier to combine terms later on. This transformation is crucial for consolidating multiple logarithmic terms into a single one, which is often the desired outcome in many mathematical problems.

Product Rule

The product rule states that $\log_b(xy) = \log_b(x) + \log_b(y)$ . This means that the logarithm of a product is the sum of the logarithms of the individual factors. For example, $\log_2(8 \times 4)$ can be rewritten as $\log_2(8) + \log_2(4)$ . This rule is invaluable for combining separate logarithmic terms into a single logarithm when those terms are added together. By applying the product rule, we can effectively merge multiple logarithms into one, simplifying the expression and making it easier to work with. Understanding and applying this rule correctly is essential for any task involving logarithmic simplification and equation solving.

Quotient Rule

The quotient rule is another crucial property, which states that $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$ . This rule tells us that the logarithm of a quotient is the difference between the logarithms of the numerator and the denominator. Consider the expression $\log_5(\frac{25}{5})$ . We can rewrite this using the quotient rule as $\log_5(25) - \log_5(5)$ . The quotient rule is particularly helpful when dealing with logarithmic expressions involving division. It allows us to separate a single logarithm into two, or conversely, combine two logarithms (subtracted from each other) into a single one. This capability is vital for simplifying complex expressions and solving logarithmic equations efficiently. Recognizing when and how to apply this rule can significantly streamline the problem-solving process.

Applying the Rules

Now, let's apply these rules to the given expression: $3 \log _5 x+4 \log _5 y$ . Our goal is to write this as a single logarithm. The first thing we'll tackle is the coefficients in front of the logarithms. We have a 3 multiplying $\log_5 xand a4 multiplying $\log_5 y. Using the power rule, we can rewrite these terms. Remember, the power rule states that $p \log_b(x) = \log_b(x^p)$ . This rule allows us to move the coefficients as exponents inside the logarithms. Applying this to our expression, we get:

$3 \log _5 x = \log _5 (x^3)$

and

$4 \log _5 y = \log _5 (y^4)$

So, our expression now looks like this:

$\log _5 (x^3) + \log _5 (y^4)$

Notice that we've successfully eliminated the coefficients by using the power rule. This is a crucial step in combining multiple logarithms into a single one. Next, we'll focus on combining the two logarithmic terms that are now added together. This is where the product rule comes into play.

The product rule states that $\log_b(x) + \log_b(y) = \log_b(xy)$ . This means that the sum of two logarithms with the same base can be written as the logarithm of the product of their arguments. In our case, we have $\log _5 (x^3) + \log _5 (y^4)$ . Applying the product rule, we combine these two logarithms into a single logarithm:

$\log _5 (x^3) + \log _5 (y^4) = \log _5 (x^3 y^4)$

Therefore, the expression $3 \log _5 x+4 \log _5 y$ can be written as a single logarithm: $\log _5 (x^3 y^4)$ . We have successfully used the power rule and the product rule to simplify the expression and achieve our goal. This step-by-step approach is key to solving similar problems efficiently and accurately. Remember, breaking down the problem into smaller, manageable steps makes the entire process much clearer and less intimidating.

Step-by-Step Solution

Let's recap the steps we took to solve this problem. This step-by-step approach will help solidify your understanding and make it easier to tackle similar problems in the future. Breaking down the solution into clear, concise steps is crucial for mastering logarithmic expressions. By following a structured approach, you can avoid common errors and build confidence in your problem-solving abilities.

Identify the logarithmic properties needed: The first step is always to recognize which logarithmic properties are relevant to the problem. In this case, we identified the power rule and the product rule as the key properties we would need to use. This initial assessment helps to create a roadmap for solving the problem, guiding you toward the appropriate techniques.
Apply the power rule: We started by using the power rule to move the coefficients in front of the logarithms as exponents. The power rule states that $p \log_b(x) = \log_b(x^p)$ . Applying this rule to $3 \log _5 x$ gave us $\log _5 (x^3)$ , and applying it to $4 \log _5 y$ gave us $\log _5 (y^4)$ . This step is crucial for eliminating coefficients and preparing the expression for further simplification.
Apply the product rule: Next, we used the product rule to combine the two logarithmic terms into a single logarithm. The product rule states that $\log_b(x) + \log_b(y) = \log_b(xy)$ . Applying this rule to $\log _5 (x^3) + \log _5 (y^4)$ gave us $\log _5 (x^3 y^4)$ . This step combines the individual logarithms into a single term, achieving the goal of writing the expression as a single logarithm.
Final Answer: By following these steps, we successfully rewrote the expression $3 \log _5 x+4 \log _5 y$ as a single logarithm: $\log _5 (x^3 y^4)$ . This final answer represents the simplified form of the original expression, making it easier to work with in subsequent calculations or applications.

Common Mistakes to Avoid

When working with logarithms, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and improve your accuracy. Avoiding common errors is just as important as understanding the rules themselves. By recognizing these mistakes, you can develop a more cautious and methodical approach to solving logarithmic problems.

Incorrect Application of the Power Rule

One frequent mistake is misapplying the power rule. Remember, the power rule applies only when the entire argument of the logarithm is raised to a power, or when there is a coefficient multiplying the entire logarithm. For instance, $2 \log_b(x)$ is not the same as $\log_b(2x)$ . The correct transformation for $2 \log_b(x)$ is $\log_b(x^2)$ . Make sure you apply the power rule only when the coefficient multiplies the entire logarithm, not just a part of the argument. This distinction is crucial for maintaining the integrity of the expression and arriving at the correct solution.

Mixing Up Product and Quotient Rules

Another common error is confusing the product and quotient rules. The product rule states that $\log_b(x) + \log_b(y) = \log_b(xy)$ , while the quotient rule states that $\log_b(x) - \log_b(y) = \log_b(\frac{x}{y})$ . It’s important to remember that addition inside the logarithm corresponds to multiplication when combining logarithms, and subtraction corresponds to division. Pay close attention to the signs and operations involved to avoid this common mix-up. A simple way to remember this is to associate addition with multiplication (both increase quantities) and subtraction with division (both decrease quantities).

Ignoring the Base

Forgetting to keep the base consistent is another significant mistake. When using logarithmic properties, the logarithms must have the same base. You cannot directly combine $\log_2(x)$ and $\log_3(y)$ using the product or quotient rule because they have different bases. If the bases are different, you might need to use the change of base formula to express the logarithms in the same base before combining them. Always double-check that the bases are the same before applying any logarithmic properties to ensure your calculations are accurate. Ignoring this step can lead to incorrect simplifications and ultimately, wrong answers.

Practice Problems

To solidify your understanding, let's look at a few practice problems. Working through examples is a great way to reinforce the concepts and build your problem-solving skills. Practice problems are essential for mastering any mathematical concept. By applying the rules and techniques you've learned to a variety of examples, you'll develop a deeper understanding and improve your ability to solve similar problems independently.

Rewrite the expression $2 \log _3 x-3 \log _3 y$ as a single logarithm.
Express $4 \log _2 a+5 \log _2 b-2 \log _2 c$ as a single logarithm.
Combine $ \frac{1}{2} \log _4 x+3 \log _4 y$ into a single logarithm.

These practice problems cover the key concepts discussed and will give you an opportunity to apply the power rule, product rule, and quotient rule. Take your time, work through each step carefully, and check your answers. Remember, the more you practice, the more comfortable and confident you'll become with logarithmic expressions.

By working through these examples, you'll reinforce your understanding of the logarithmic properties and develop the skills needed to solve more complex problems. Keep practicing, and you'll become a pro at manipulating logarithmic expressions!

In conclusion, mastering logarithmic properties is essential for simplifying and combining logarithmic expressions. By understanding and applying the power rule, product rule, and quotient rule, you can effectively rewrite multiple logarithms as a single logarithm. Remember to avoid common mistakes and practice regularly to build your skills and confidence. Keep up the great work, and you'll be a logarithm expert in no time!