Condensing Logarithmic Expressions: Step-by-Step Guide

Hey math enthusiasts! Today, we're diving deep into the world of logarithms and learning how to condense multiple logarithmic terms into a single, elegant expression. This skill is super useful in various mathematical contexts, from solving equations to simplifying complex formulas. So, let's get started and break down the process step by step.

Understanding Logarithmic Properties

Before we jump into the problem, let's quickly recap the key logarithmic properties we'll be using. These properties are the foundation of condensing and simplifying logarithmic expressions:

1. Power Rule: This rule states that $\log_b(x^p) = p \log_b(x)$. In simpler terms, an exponent inside the logarithm can be brought down as a coefficient, and vice versa. This is perhaps the most crucial rule when dealing with coefficients in front of logarithms. It allows us to rewrite terms like $5 \log_7 t$ as $\log_7 (t^5)$. Mastering this rule is essential for both expanding and condensing logarithmic expressions. It's like having a superpower that transforms the structure of our logarithmic terms, making them easier to manipulate.

2. Product Rule: This rule tells us that $\log_b(xy) = \log_b(x) + \log_b(y)$. The logarithm of a product is equal to the sum of the logarithms. Think of it as a way to combine separate logarithmic terms into a single logarithm when they are added together. For instance, if we have $\log_7 (t^5) + \log_7 (u^9)$, we can combine them into a single logarithm: $\log_7 (t^5 u^9)$. Understanding when and how to apply this rule can significantly simplify complex expressions.

3. Quotient Rule: This rule states that $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$. The logarithm of a quotient is equal to the difference of the logarithms. It's the counterpart to the product rule and comes into play when we have logarithmic terms being subtracted. For example, if we see $\log_7 (t^5) - \log_7 (u^9)$, we can rewrite it as $\log_7 (\frac{t^5}{u^9})$. Recognizing this pattern is key to efficiently condensing logarithmic expressions.

With these rules in our toolkit, we're well-equipped to tackle the given problem!

Problem Breakdown: Condensing the Logarithmic Expression

Our mission is to condense the expression: $5 \log _7 t-9 \log _7 u-8 \log _7 v$ into a single logarithm with a coefficient of 1. Let's walk through the process step-by-step, applying the logarithmic properties we just discussed.

Step 1: Applying the Power Rule

The first thing we notice is that we have coefficients in front of our logarithms. Remember the power rule? It's our go-to tool for dealing with these coefficients. We'll use the power rule to move the coefficients as exponents inside the logarithms:

• $5 \log _7 t$ becomes $\log _7 (t^5)$
• $9 \log _7 u$ becomes $\log _7 (u^9)$
• $8 \log _7 v$ becomes $\log _7 (v^8)$

Now our expression looks like this:

$$\log _7 (t^5) - \log _7 (u^9) - \log _7 (v^8)$$

This transformation is a crucial first step because it gets rid of the coefficients, allowing us to use the product and quotient rules more effectively. By converting the coefficients into exponents, we're setting the stage for combining the logarithmic terms into a single logarithm.

Step 2: Applying the Quotient Rule (Part 1)

Next, we see subtraction between the logarithmic terms. Subtraction hints at the quotient rule. Let's apply the quotient rule to the first two terms:

$$\log _7 (t^5) - \log _7 (u^9) = \log _7 (\frac{t^5}{u^9})$$

Now our expression is:

$$\log _7 (\frac{t^5}{u^9}) - \log _7 (v^8)$$

We've successfully combined the first two terms into a single logarithm. This simplification is a significant step forward, as it reduces the number of logarithmic terms we're dealing with. By applying the quotient rule, we're effectively merging two logarithmic expressions into one, making the overall expression more manageable.

Step 3: Applying the Quotient Rule (Part 2)

We still have subtraction, so let's apply the quotient rule again. This time, we'll combine the remaining two terms:

$$\log _7 (\frac{t^5}{u^9}) - \log _7 (v^8) = \log _7 (\frac{t^5}{u^9} \div v^8)$$

To simplify the division, we can rewrite dividing by $v^8$ as multiplying by $\frac{1}{v^8}$:

$$\log _7 (\frac{t^5}{u^9} \times \frac{1}{v^8}) = \log _7 (\frac{t^5}{u^9 v^8})$$

We've now condensed the entire expression into a single logarithm!

Final Answer: The Condensed Logarithmic Expression

After applying the power and quotient rules, we've successfully condensed the given logarithmic expression into a single logarithm with a coefficient of 1. The simplified expression is:

$$\log _7 (\frac{t^5}{u^9 v^8})$$

So, the correct answer is: $\log _7(\frac{t^5}{u9 v^8})$.

Key Takeaways and Practice

Condensing logarithmic expressions might seem tricky at first, but with practice, it becomes second nature. The key is to remember and apply the logarithmic properties systematically. Always start by using the power rule to eliminate coefficients, then use the product rule for addition and the quotient rule for subtraction.

Remember, math is a skill best honed through practice. Try tackling more problems like this, and you'll become a pro at manipulating logarithmic expressions in no time! Keep practicing, and you'll find these concepts becoming clearer and more intuitive.

Here are some additional tips to keep in mind:

• Pay attention to the base: All logarithms in the expression must have the same base to apply these rules.
• Work step-by-step: Don't try to do everything at once. Break the problem down into smaller, manageable steps.
• Double-check your work: It's easy to make a small mistake, so take a moment to review each step.

By following these guidelines and practicing consistently, you'll be well on your way to mastering logarithmic expressions. Happy calculating, guys!