Condense Logarithmic Expressions: A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of logarithms. Our mission? To take a somewhat complex logarithmic expression and condense it into a single, neat logarithm. Specifically, we're tackling the expression: $2 \log _7 w+3(\log _7 y-5 \log _7 z)$. Buckle up, because we're about to make this look easy!

Understanding the Basics of Logarithms

Before we jump into the problem, let's quickly recap what logarithms are all about. A logarithm is essentially the inverse operation to exponentiation. When we write $\log_b a = c$, it means that $b^c = a$. In simpler terms, the logarithm tells you what exponent you need to raise the base (b) to, in order to get a certain number (a). The base of our logarithms in this problem is 7, so we're dealing with $\log_7$.

Key Properties of Logarithms:

To successfully condense our expression, we need to remember a few key properties of logarithms. These properties are the tools in our toolbox that will help us simplify and combine the logarithmic terms.

1. Power Rule: $\log_b (x^p) = p \log_b (x)$. This rule tells us that if you have a logarithm of a number raised to a power, you can bring the power down and multiply it by the logarithm.
2. Product Rule: $\log_b (xy) = \log_b (x) + \log_b (y)$. This rule states that the logarithm of a product is the sum of the logarithms.
3. Quotient Rule: $\log_b (\frac{x}{y}) = \log_b (x) - \log_b (y)$. This rule says that the logarithm of a quotient is the difference of the logarithms.
4. Constant Multiple Rule: $c \log_b(x) = \log_b(x^c)$. This is a variation of the power rule and is useful when you need to move a constant factor into the logarithm as an exponent.

Make sure you have a solid grasp of these properties, as they are the foundation for manipulating and simplifying logarithmic expressions. Keep these rules handy as we proceed through our problem.

Step-by-Step Solution

Now, let's break down the expression $2 \log _7 w+3(\log _7 y-5 \log _7 z)$ step-by-step.

Step 1: Apply the Power Rule

Our first goal is to deal with the coefficients in front of the logarithms. We can use the power rule to move these coefficients as exponents inside the logarithms.

• For the first term, $2 \log_7 w$, we apply the power rule to get $\log_7 (w^2)$.
• For the term inside the parenthesis, $5 \log_7 z$, we get $\log_7 (z^5)$.

So, our expression now looks like this: $\log_7 (w^2) + 3(\log_7 y - \log_7 (z^5))$.

Step 2: Simplify Inside the Parentheses

Next, let's simplify the expression inside the parentheses. We have $\log_7 y - \log_7 (z^5)$. This is a difference of logarithms, so we can use the quotient rule to combine them.

Using the quotient rule, $\log_7 y - \log_7 (z^5)$ becomes $\log_7 (\frac{y}{z^5})$.

Now, our expression is: $\log_7 (w^2) + 3 \log_7 (\frac{y}{z^5})$.

Step 3: Apply the Power Rule Again

We still have a coefficient in front of the second logarithm, namely the '3'. We'll use the power rule once more to move this coefficient inside as an exponent.

So, $3 \log_7 (\frac{y}{z^5})$ becomes $\log_7 ((\frac{y}{z^5})^3)$.

This simplifies to $\log_7 (\frac{y^3}{z^{15}})$.

Now, the expression looks like this: $\log_7 (w^2) + \log_7 (\frac{y^3}{z^{15}})$.

Step 4: Combine the Logarithms

We now have a sum of two logarithms with the same base. We can use the product rule to combine them into a single logarithm.

Using the product rule, $\log_7 (w^2) + \log_7 (\frac{y^3}{z^{15}})$ becomes $\log_7 (w^2 \cdot \frac{y^3}{z^{15}})$.

This simplifies to $\log_7 (\frac{w^2 y^3}{z^{15}})$.

Final Answer:

Therefore, the expression $2 \log _7 w+3(\log _7 y-5 \log _7 z)$ can be written as a single logarithm: \log_7 (\frac{w^2 y^3}{z^{15}}}).

Common Mistakes to Avoid

• Forgetting the Power Rule: Always remember to deal with coefficients by moving them as exponents first.
• Incorrectly Applying the Quotient Rule: Make sure you subtract the logarithms in the correct order when using the quotient rule.
• Mixing Up the Product and Quotient Rules: Remember that addition turns into multiplication inside the logarithm, and subtraction turns into division.
• Not Distributing Properly: If there are parentheses and coefficients outside, make sure you distribute correctly, especially when subtraction is involved.

Practice Problems

To really nail this down, try these practice problems:

1. Simplify: $3 \log_2 x + 2(\log_2 y - 4 \log_2 z)$
2. Simplify: $\frac{1}{2} \log_5 a - 3 \log_5 b + \log_5 c$
3. Simplify: $4 \log_3 m - 2(\log_3 n + 5 \log_3 p)$

Work through these problems, and you'll become a pro at condensing logarithmic expressions in no time! Always remember to apply the power, product, and quotient rules correctly.

Conclusion

So, there you have it! Condensing logarithmic expressions might seem tricky at first, but with a solid understanding of the logarithm properties and a step-by-step approach, you can simplify even the most complex expressions. Remember to practice regularly, and you'll master this skill in no time. Keep up the great work, guys, and happy logarithm-ing!