Logarithm Calculation: Find The Value Of Log Expression

by Andrew McMorgan 56 views

Hey guys! Today, we're diving into the fascinating world of logarithms with a problem that might look a bit intimidating at first, but trust me, it’s totally manageable. We're going to break down this logarithm problem step by step, making sure everyone understands the process. So, grab your thinking caps, and let’s get started!

Understanding the Problem

The problem we're tackling involves finding the value of a logarithmic expression, given some initial logarithmic values. Specifically, we are given that log⁡ax=3{\log_a x = 3}, log⁡ay=4{\log_a y = 4}, and log⁡az=5{\log_a z = 5}. Our mission, should we choose to accept it (and we do!), is to find the value of log⁡ay3z44x4z−44{\log_a \frac{\sqrt[4]{y^3 z^4}}{\sqrt[4]{x^4 z^{-4}}}}. This looks like a handful, right? But don't worry, we'll use the properties of logarithms and exponents to simplify it. We'll convert those nasty roots into fractional exponents, and then strategically apply the logarithmic rules to dissect the expression. It's like performing surgery on a math problem – precise, calculated, and ultimately satisfying when you get the right answer. Remember, the key to logarithms is understanding their properties and applying them systematically. So let's roll up our sleeves and dive in!

Breaking Down the Expression

Okay, let's get our hands dirty and start dissecting this beast of an expression. The first thing we need to do is rewrite the radicals using fractional exponents. Remember, a fourth root is the same as raising something to the power of 14{\frac{1}{4}}. So, let’s rewrite the expression inside the logarithm:

y3z44x4z−44=(y3z4)14(x4z−4)14{\frac{\sqrt[4]{y^3 z^4}}{\sqrt[4]{x^4 z^{-4}}} = \frac{(y^3 z^4)^{\frac{1}{4}}}{(x^4 z^{-4})^{\frac{1}{4}}}}

Now, we need to distribute the exponent of 14{\frac{1}{4}} to each term inside the parentheses. This is just a basic rule of exponents, but it's super important to get it right. When we raise a power to a power, we multiply the exponents. So, let's do that:

=y34z44x44z−44=y34z1x1z−1{= \frac{y^{\frac{3}{4}} z^{\frac{4}{4}}}{x^{\frac{4}{4}} z^{-\frac{4}{4}}} = \frac{y^{\frac{3}{4}} z^1}{x^1 z^{-1}}}

See how much cleaner that looks already? We've gotten rid of the radicals and now we just have fractional and integer exponents. But we can simplify this even further. Remember, when you divide terms with the same base, you subtract the exponents. So, let's move that z−1{z^{-1}} from the denominator to the numerator by changing the sign of its exponent:

=y34z1x−1z1=y34x−1z2{= y^{\frac{3}{4}} z^1 x^{-1} z^1 = y^{\frac{3}{4}} x^{-1} z^2}

We've now got a much simpler expression: y34x−1z2{y^{\frac{3}{4}} x^{-1} z^2}. This is the heart of the problem, and now we can apply the logarithmic properties with confidence. You might be thinking, "Wow, that was a lot of exponent manipulation!" And you're right, it was. But that's often the name of the game with these kinds of problems. Simplifying the expression using exponent rules makes the logarithmic part much easier to handle. So, take a deep breath, appreciate the progress we've made, and let’s move on to the logarithmic part!

Applying Logarithmic Properties

Alright, now comes the fun part – applying those sweet, sweet logarithmic properties! We've successfully simplified our expression to y34x−1z2{y^{\frac{3}{4}} x^{-1} z^2}. Now we need to take the logarithm base a{a} of this whole thing. So, we're looking at:

log⁡a(y34x−1z2){\log_a (y^{\frac{3}{4}} x^{-1} z^2)}

Remember those log properties we talked about earlier? This is where they shine. The first property we'll use is the product rule, which states that the logarithm of a product is the sum of the logarithms. This means we can break up the logarithm of our product into a sum of individual logarithms:

log⁡a(y34x−1z2)=log⁡a(y34)+log⁡a(x−1)+log⁡a(z2){\log_a (y^{\frac{3}{4}} x^{-1} z^2) = \log_a (y^{\frac{3}{4}}) + \log_a (x^{-1}) + \log_a (z^2)}

Awesome! We've separated our complex logarithm into three simpler ones. Now, we're going to use another crucial property: the power rule. This rule says that the logarithm of something raised to a power is the power times the logarithm of the something. In other words, we can bring those exponents out front as coefficients:

=34log⁡ay+(−1)log⁡ax+2log⁡az{= \frac{3}{4} \log_a y + (-1) \log_a x + 2 \log_a z}

Look at that! We've transformed our expression into a linear combination of logarithms. We've gone from a complex fraction with radicals to a simple sum of terms, each involving a logarithm we know the value of. This is the power of logarithmic properties in action. It’s like turning a complicated puzzle into a set of easy-to-fit pieces. Now, all that's left is to substitute the values we were given at the beginning of the problem. Get ready for the final step, guys!

Substituting Values and Calculating

Okay, we've reached the home stretch! We've simplified our logarithmic expression to:

34log⁡ay+(−1)log⁡ax+2log⁡az{\frac{3}{4} \log_a y + (-1) \log_a x + 2 \log_a z}

And remember, at the very beginning of the problem, we were given the values: log⁡ax=3{\log_a x = 3}, log⁡ay=4{\log_a y = 4}, and log⁡az=5{\log_a z = 5}. Now it's just a matter of plugging these values into our expression and doing some basic arithmetic. This is the satisfying part where all our hard work pays off!

Let's substitute those values in:

=34(4)+(−1)(3)+2(5){= \frac{3}{4} (4) + (-1) (3) + 2 (5)}

Now, let's do the multiplication:

=3−3+10{= 3 - 3 + 10}

And finally, let's add it all up:

=10{= 10}

Boom! We've got our answer. The value of log⁡ay3z44x4z−44{\log_a \frac{\sqrt[4]{y^3 z^4}}{\sqrt[4]{x^4 z^{-4}}}} is 10. How cool is that? We took a complicated-looking problem, broke it down using the properties of exponents and logarithms, and arrived at a neat, clean solution. Give yourselves a pat on the back, guys. You've earned it!

Final Answer

So, to recap, we started with a complex logarithmic expression involving radicals and fractions. We used the properties of exponents to simplify the expression inside the logarithm, then applied the product and power rules of logarithms to break the problem down further. Finally, we substituted the given values and performed the arithmetic to arrive at the final answer. The final answer is:

log⁡ay3z44x4z−44=10{\log_a \frac{\sqrt[4]{y^3 z^4}}{\sqrt[4]{x^4 z^{-4}}} = 10}

This problem is a perfect example of how understanding the fundamental properties of logarithms and exponents can help you tackle even the most intimidating-looking problems. Remember, math isn't about memorizing formulas; it's about understanding the underlying concepts and applying them creatively. Keep practicing, keep exploring, and keep having fun with math! You've got this!