Expanding Logarithmic Expressions A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of logarithms and how to expand them using their awesome properties. We'll specifically tackle the expression $\log \sqrt{\frac{x y^7}{z^3}}$ and break it down step-by-step until each logarithm involves only one variable, without any radicals or exponents. This is a super useful skill in mathematics, especially when dealing with complex equations or simplifying expressions in calculus and other advanced topics. So, buckle up and let's get started!

Understanding the Properties of Logarithms

Before we jump into the problem, let's quickly recap the key properties of logarithms that we'll be using. These properties are the foundation for expanding and simplifying logarithmic expressions. Think of them as the essential tools in our logarithm toolbox.

1. Product Rule: This rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, it's expressed as:

   $\log_b (MN) = \log_b M + \log_b N$

   In simpler terms, if you're taking the logarithm of something that's being multiplied, you can split it into the sum of the logarithms of each part.

2. Quotient Rule: This rule is similar to the product rule, but it deals with division. It states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. The formula looks like this:

   $\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N$

   Basically, if you're taking the logarithm of a fraction, you can split it into the logarithm of the top minus the logarithm of the bottom.

3. Power Rule: This rule is super handy for dealing with exponents inside logarithms. It states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. The formula is:

   $\log_b (M^p) = p \log_b M$

   So, if you have an exponent inside a logarithm, you can bring it out front as a multiplier.

These three properties – the product rule, the quotient rule, and the power rule – are our main weapons in this logarithmic expansion adventure. Mastering them will make expanding and simplifying logarithmic expressions a breeze. We'll be using these properties in combination to tackle our main problem, so make sure you've got them down!

Breaking Down the Expression: $\log \sqrt{\frac{x y^7}{z^3}}$

Okay, let's get our hands dirty with the actual expression: $\log \sqrt{\frac{x y^7}{z^3}}$. Our goal is to expand this logarithm so that each term involves only one variable and has no radicals or exponents. We'll do this systematically, one step at a time, using the properties of logarithms we just discussed. It's like peeling an onion, layer by layer, to reveal the simpler components inside.

Step 1: Dealing with the Square Root

The first thing we notice is the square root. Remember that a square root is the same as raising something to the power of $\frac{1}{2}$. So, we can rewrite the expression as:

$\log \left(\frac{x y^7}{z^3}\right)^{\frac{1}{2}}$

Now, we can use the power rule to bring the exponent $\frac{1}{2}$ out front:

$\frac{1}{2} \log \left(\frac{x y^7}{z^3}\right)$

This step is crucial because it gets rid of the radical, making the expression much easier to work with. We've effectively moved the square root from being "inside" the logarithm to being a multiplier outside.

Step 2: Applying the Quotient Rule

Next up, we have a fraction inside the logarithm. This is where the quotient rule comes into play. Remember, the quotient rule states that the logarithm of a fraction is the logarithm of the numerator minus the logarithm of the denominator. Applying this rule, we get:

$\frac{1}{2} \left[ \log (x y^7) - \log (z^3) \right]$

Notice the brackets here – they're super important! The $\frac{1}{2}$ multiplies the entire expression that results from expanding the logarithm, so we need to keep track of that. We've now separated the fraction into two separate logarithmic terms, which is a big step forward.

Step 3: Using the Product Rule

Now, let's focus on the first term inside the brackets: $\log (x y^7)$. We have a product here, so we can use the product rule. The product rule tells us that the logarithm of a product is the sum of the logarithms of the individual factors. So, we can rewrite $\log (x y^7)$ as:

$\log x + \log y^7$

Substituting this back into our expression, we get:

$\frac{1}{2} \left[ \log x + \log y^7 - \log (z^3) \right]$

We're getting closer and closer to our goal! We've broken down the product in the numerator, and we're left with individual terms that are much simpler.

Step 4: Applying the Power Rule Again

We still have exponents inside the logarithms in the terms $\log y^7$ and $\log (z^3)$. This is another perfect opportunity to use the power rule. We can bring the exponents 7 and 3 out front as multipliers:

$\log y^7 = 7 \log y$

$\log z^3 = 3 \log z$

Substituting these back into our expression, we now have:

$\frac{1}{2} \left[ \log x + 7 \log y - 3 \log z \right]$

Step 5: Distributing the $\frac{1}{2}$

Finally, we need to distribute the $\frac{1}{2}$ that's multiplying the entire expression. This means multiplying each term inside the brackets by $\frac{1}{2}$:

$\frac{1}{2} \log x + \frac{7}{2} \log y - \frac{3}{2} \log z$

And there you have it! We've successfully expanded the original logarithmic expression into a sum and difference of logarithms, each involving only one variable and without any radicals or exponents. It might seem like a lot of steps, but by applying the properties of logarithms systematically, we were able to break down a complex expression into its simpler components.

The Expanded Form: $\frac{1}{2} \log x + \frac{7}{2} \log y - \frac{3}{2} \log z$

So, the fully expanded form of the expression $\log \sqrt{\frac{x y^7}{z^3}}$ is:

$\frac{1}{2} \log x + \frac{7}{2} \log y - \frac{3}{2} \log z$

This is our final answer! Notice how each term now involves only one variable ($x$, $y$, or $z$), and there are no more radicals or exponents inside the logarithms. We've achieved our goal by strategically applying the properties of logarithms.

Why is Expanding Logarithms Important?

You might be wondering,