Condense Logarithmic Expressions: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of logarithms to tackle a common challenge: condensing multiple logarithmic expressions into a single, neat logarithm. Specifically, we're going to break down how to simplify an expression like $11 \log _2(w)+3 \log _2(y)-7 \log _2(z)$. Trust me, it's not as intimidating as it looks! By the end of this guide, you'll be able to handle these types of problems with confidence. So, let's get started and make logarithms a little less mysterious.

Understanding the Basics of Logarithms

Before we jump into condensing logarithmic expressions, let's quickly review some fundamental properties of logarithms. Understanding these properties is crucial for simplifying and manipulating logarithmic expressions effectively. Logarithms are essentially the inverse operation of exponentiation. In other words, if we have an expression like $b^x = y$, we can rewrite it in logarithmic form as $\log_b(y) = x$. Here, $b$ is the base of the logarithm, $y$ is the argument (the value we're taking the logarithm of), and $x$ is the exponent to which we must raise $b$ to obtain $y$. Key properties that we'll use include:

• Product Rule: $\log_b(mn) = \log_b(m) + \log_b(n)$. This rule states that the logarithm of a product is the sum of the logarithms of the individual factors.
• Quotient Rule: $\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)$. This rule states that the logarithm of a quotient is the difference between the logarithms of the numerator and the denominator.
• Power Rule: $\log_b(m^p) = p \log_b(m)$. This rule states that the logarithm of a number raised to a power is the product of the power and the logarithm of the number.

These properties are the building blocks for simplifying and condensing logarithmic expressions. Mastering these rules will make the process much smoother and more intuitive. For example, when we see a sum of logarithms, we should immediately think of the product rule and consider combining the arguments into a single logarithm. Similarly, a difference of logarithms suggests using the quotient rule. And, most importantly for our problem today, coefficients in front of logarithms can be handled using the power rule to bring them inside as exponents. With these basics in mind, let's move on to the specific problem we want to solve.

Applying the Power Rule

The expression we want to condense is $11 \log _2(w)+3 \log _2(y)-7 \log _2(z)$. The first step in condensing this logarithmic expression is to apply the power rule. Remember, the power rule states that $\log_b(m^p) = p \log_b(m)$. In our expression, we have coefficients in front of each logarithm: 11, 3, and -7. We can use the power rule to bring these coefficients inside the logarithms as exponents. Applying the power rule to each term, we get:

$11 \log _2(w) = \log _2(w^{11})$

$3 \log _2(y) = \log _2(y^3)$

$-7 \log _2(z) = \log _2(z^{-7})$

Now our expression looks like this:

$\log _2(w^{11}) + \log _2(y^3) - \log _2(z^{-7})$

By applying the power rule, we've eliminated the coefficients and transformed the expression into a sum and difference of logarithms. This sets us up perfectly for the next step, where we'll use the product and quotient rules to combine these logarithms into a single logarithm. Remember, the key is to take it step by step, applying one rule at a time until we reach our goal. This approach makes the process much more manageable and reduces the chance of making errors. The power rule is often the first rule to apply when condensing logarithms, especially when you have coefficients in front of the logarithmic terms.

Using the Product Rule to Combine Logarithms

Now that we've applied the power rule and have the expression $\log _2(w^{11}) + \log _2(y^3) - \log _2(z^{-7})$, we can move on to using the product rule to combine the logarithms that are being added. The product rule states that $\log_b(mn) = \log_b(m) + \log_b(n)$. In our expression, we have $\log _2(w^{11}) + \log _2(y^3)$, which can be combined into a single logarithm using the product rule:

$\log _2(w^{11}) + \log _2(y^3) = \log _2(w^{11}y^3)$

Now our expression looks like this:

$\log _2(w^{11}y^3) - \log _2(z^{-7})$

We've successfully combined the first two logarithmic terms into a single logarithm. This simplifies the expression and brings us one step closer to our final goal of expressing the entire expression as a single logarithm. The product rule is a powerful tool for combining logarithms, and it's essential to recognize when it can be applied. Look for sums of logarithms with the same base, and you can usually combine them into a single logarithm of the product of their arguments. Next, we'll handle the subtraction of logarithms using the quotient rule.

Applying the Quotient Rule to Finalize the Single Logarithm

With our expression now simplified to $\log _2(w^{11}y^3) - \log _2(z^{-7})$, we're ready to apply the quotient rule. The quotient rule states that $\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)$. In our case, we have a difference of two logarithms, so we can combine them into a single logarithm using the quotient rule:

$\log _2(w^{11}y^3) - \log _2(z^{-7}) = \log _2(\frac{w^{11}y^3}{z^{-7}})$

To simplify further, remember that dividing by $z^{-7}$ is the same as multiplying by $z^7$. Therefore, we can rewrite the expression as:

$\log _2(\frac{w^{11}y^3}{z^{-7}}) = \log _2(w^{11}y^3z^7)$

And that's it! We've successfully condensed the original expression into a single logarithm: $\log _2(w^{11}y^3z^7)$.

Final Answer

So, the final answer is:

$\log _2(w^{11}y^3z^7)$

By systematically applying the power rule, product rule, and quotient rule, we were able to condense the given logarithmic expression into a single logarithm. Remember, the key to success with these types of problems is to break them down into smaller, manageable steps. Start by applying the power rule to eliminate any coefficients, then use the product rule to combine sums of logarithms, and finally, use the quotient rule to combine differences of logarithms. With practice, you'll become more comfortable with these properties and be able to condense logarithmic expressions with ease. Keep practicing, and you'll master these logarithmic manipulations in no time!