Hey guys! Today, we're diving into the world of logarithms and tackling a common challenge: expressing logarithmic expressions as a single logarithm and simplifying them. This is a crucial skill in mathematics, especially when dealing with exponential and logarithmic equations. We'll break down the process step by step, making it super easy to understand. So, let's get started!
Understanding Logarithmic Properties
Before we jump into the problem, let's quickly review the key logarithmic properties that we'll be using. These properties are the foundation of simplifying logarithmic expressions. Remember, logarithms are essentially the inverse of exponential functions, and these properties reflect that relationship. Mastering these properties is key to effectively manipulating logarithmic expressions. Let's refresh our memory with the following:
- 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: $\ln(AB) = \ln(A) + \ln(B)$. Think of it as a way to break down a complex logarithm into simpler parts. For instance, if you have $\ln(6)$, you can rewrite it as $\ln(2 \cdot 3)$, which then becomes $\ln(2) + \ln(3)$. This can be incredibly useful when dealing with expressions that involve multiplication inside the logarithm.
- Quotient Rule: This rule is the counterpart to the product rule and 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 is: $\ln(\frac{A}{B}) = \ln(A) - \ln(B)$. Imagine you're working with $\ln(\frac{10}{2})$. You can transform this into $\ln(10) - \ln(2)$. This rule is particularly helpful when you have fractions within your logarithmic expressions.
- Power Rule: This rule is perhaps one of the most frequently used in simplifying 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: $\ln(A^p) = p \ln(A)$. This is where things get really interesting. Suppose you encounter $\ln(x^5)$. Using the power rule, you can rewrite it as $5 \ln(x)$. The power rule allows us to move exponents outside the logarithm, which is a game-changer when it comes to simplification. This rule is crucial for handling expressions where the argument of the logarithm has an exponent.
These three properties – the product rule, quotient rule, and power rule – are the fundamental tools in our logarithmic toolkit. Understanding when and how to apply them is the key to simplifying and manipulating logarithmic expressions. Now that we've refreshed these properties, let's move on to tackling the specific problem at hand. We'll see how these rules come into play as we express the given expression as a single logarithm and simplify it.
Problem Statement: $\ln x^3-5 \ln \sqrt[5]{x}$
Okay, let's jump right into the problem we have: $\ln x^3 - 5 \ln \sqrt[5]{x}$. Our mission, should we choose to accept it (and we do!), is to express this as a single logarithm and then simplify it as much as possible. This type of problem often appears in algebra and calculus, so mastering it is a fantastic investment in your math skills. To begin, we'll leverage the logarithmic properties we just discussed to condense and simplify this expression.
First, we have the term $\ln x^3$. Notice that this term is already in a relatively simple form. It has a power within the logarithm, which is a great opportunity to use the power rule. We'll keep this in mind for our next step. The second term, $5 \ln \sqrt[5]{x}$, looks a bit more complex. We have a constant multiplied by a logarithm, and the argument of the logarithm is a radical. Radicals can sometimes be tricky, but they're nothing we can't handle. Remember that a radical can be expressed as a fractional exponent. So, the fifth root of $x$, which is $\sqrt[5]{x}$, can be rewritten as $x^{\frac{1}{5}}$. This transformation is key because it allows us to apply the power rule more easily. By converting the radical to a fractional exponent, we've set ourselves up for the next simplification step. This is a common strategy in simplifying logarithmic expressions: look for opportunities to rewrite radicals as exponents. Once we've made this transformation, the expression starts to look much more manageable. We've essentially prepared the ground for using the power rule, which will help us bring the exponent outside the logarithm and further simplify the expression. Let's move on to the next step where we'll put these transformations into action and continue our journey towards expressing this as a single, simplified logarithm. Remember, the goal is to make the expression as concise and clear as possible. So, with our logarithmic properties in hand and our initial transformations complete, we're well on our way to solving this problem.
Applying Logarithmic Properties
Now, let's roll up our sleeves and apply those logarithmic properties to our expression: $\ln x^3 - 5 \ln \sqrt[5]x}$. Remember, we've already identified that $\sqrt[5]{x}$ can be rewritten as $x^{\frac{1}{5}}$. So, let's substitute that into our expression. This gives us{5}}$. See how much cleaner that looks already? The fractional exponent is much easier to work with than the radical.
Next up, we're going to use the power rule. The power rule is our best friend when we have exponents inside a logarithm. It allows us to bring the exponent outside as a coefficient. In our expression, we have two terms where the power rule can be applied. For the first term, $\ln x^3$, the power is 3. So, applying the power rule, we get $3 \ln x$. For the second term, $5 \ln x^{\frac{1}{5}}$, we have an exponent of $\frac{1}{5}$. Applying the power rule here, we move the $\frac{1}{5}$ outside the logarithm, which gives us $5 \cdot \frac{1}{5} \ln x$. This is where things get interesting because we have a constant multiplied by a fraction. Remember, math is all about looking for these opportunities to simplify!
Let's simplify the second term further. We have $5 \cdot \frac{1}{5}$, which is simply 1. So, $5 \cdot \frac{1}{5} \ln x$ simplifies to $1 \cdot \ln x$, or just $\ln x$. Now, our entire expression looks like this: $3 \ln x - \ln x$. We've successfully applied the power rule and simplified the expression quite a bit. Notice that we now have two terms that both involve $\ln x$. This is a crucial observation because it means we can combine these terms. We're almost there! The next step is to combine these logarithmic terms, which will bring us closer to expressing the original expression as a single logarithm. By strategically using the power rule and simplifying constants, we've made the expression much more manageable and set ourselves up for the final simplification.
Combining Logarithmic Terms
Alright, we're in the home stretch! Our expression now looks like this: $3 \ln x - \ln x$. Notice anything familiar? We have two terms that are like terms – they both contain $\ln x$. This means we can combine them just like we would combine $3y - y$ in algebra. Think of $
x$ as a variable; it makes the process much clearer.
So, how do we combine $3 \ln x - \ln x$? Well, it's just like subtracting coefficients. We have 3 of something ($\ln x$) minus 1 of that same thing ($\ln x$). So, $3 \ln x - \ln x$ is equal to $2 \ln x$. See? Simple as that! We've now condensed our expression significantly. We've gone from two logarithmic terms to just one. This is a major step towards our goal of expressing the original expression as a single logarithm.
But wait, we're not quite done yet. While $2 \ln x$ is a single term, it's not in the most simplified form possible. We still have that coefficient of 2 hanging out in front of the logarithm. This is where we can use the power rule in reverse. Remember, the power rule states that $\ln(A^p) = p \ln(A)$. We used it earlier to bring exponents outside the logarithm, but we can also use it to bring coefficients inside as exponents. This is a powerful technique for further simplifying logarithmic expressions. In our case, we have $2 \ln x$. We can think of the 2 as the $p$ in the power rule formula. So, we can bring that 2 inside the logarithm as an exponent. This means $2 \ln x$ can be rewritten as $\ln x^2$. And there you have it! We've successfully expressed our original expression as a single logarithm.
Final Answer: $\ln x^2$
Drumroll, please! After all our hard work, we've arrived at the final answer. The simplified form of the expression $\ln x^3 - 5 \ln \sqrt[5]{x}$ expressed as a single logarithm is $
x^2$. How cool is that? We took a seemingly complex expression and, by strategically applying logarithmic properties, we simplified it down to a single, elegant term. This is a testament to the power of understanding and applying mathematical rules.
Let's recap the steps we took to get here. First, we rewrote the radical as a fractional exponent. This made it easier to apply the power rule. Then, we used the power rule to move the exponents outside the logarithms as coefficients. We simplified any constants and combined like terms. Finally, we used the power rule in reverse to bring the coefficient back inside the logarithm as an exponent, resulting in our final answer. This process highlights the importance of being flexible and recognizing when to apply each logarithmic property. It's like having a set of tools in a toolbox; knowing which tool to use for which job is crucial.
This type of problem is a classic example of how logarithms can be manipulated and simplified. Mastering these techniques is essential for anyone working with exponential and logarithmic functions, whether it's in algebra, calculus, or even in real-world applications like finance or engineering. So, pat yourselves on the back, guys! You've successfully navigated the world of logarithms and emerged victorious. Keep practicing, and these types of problems will become second nature. Remember, the key is to understand the properties and apply them strategically. With a little practice, you'll be simplifying logarithmic expressions like a pro!