Single Logarithm: Simplifying Log Expressions

Hey guys! Let's dive into simplifying logarithmic expressions. Today, we're tackling the expression:

$$\log _5 \sqrt{x}-\log _5 x^9$$

Our goal is to combine these logarithmic terms into a single logarithm. To do this, we'll leverage some key properties of logarithms. So, grab your thinking caps, and let's get started!

Understanding the Properties of Logarithms

Before we jump into the problem, let’s quickly recap the logarithmic properties that will help us simplify this expression. These properties are the bread and butter of logarithmic manipulation, and understanding them is crucial. These rules allow us to manipulate complex logarithmic expressions into simpler, more manageable forms.

First, there's the power rule, which states that the logarithm of a number raised to a power is equal to the power times the logarithm of the number. Mathematically, this is expressed as:

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

This rule is super handy for dealing with exponents inside logarithms. For instance, if you have something like $\log_2(8^3)$, you can rewrite it as $3 \log_2(8)$, which is much easier to handle. The power rule helps to transform exponential relationships within logarithms into multiplicative relationships, making calculations and simplifications smoother.

Next up, we have the quotient rule, which says that the logarithm of the quotient of two numbers is equal to the difference of their logarithms. In equation form:

$$\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$$

This rule is essential when you see subtraction between logarithms with the same base. It allows you to combine two separate logarithmic terms into a single logarithm of a fraction. For example, if you encounter $\log_3(27) - \log_3(9)$, you can use the quotient rule to rewrite it as $\log_3(\frac{27}{9})$, which simplifies to $\log_3(3)$. This rule is particularly useful in condensing expressions and solving logarithmic equations.

Another crucial property is the product rule, which states that the logarithm of the product of two numbers is equal to the sum of their logarithms. It’s expressed as:

$$\log_b(xy) = \log_b(x) + \log_b(y)$$

This rule complements the quotient rule and is used when you have addition between logarithms of the same base. It allows you to combine the sum of logarithms into a single logarithm of a product. For instance, $\log_5(5) + \log_5(25)$ can be rewritten using the product rule as $\log_5(5 \times 25)$, which simplifies to $\log_5(125)$. The product rule is invaluable in expanding and simplifying logarithmic expressions, especially when dealing with multiplication within logarithms.

Finally, let's not forget the fundamental identity that $\log_b(b) = 1$. This property is a cornerstone in simplifying expressions where the base of the logarithm matches the argument. For example, $\log_{10}(10)$ is simply 1, and $\log_e(e)$ (also known as the natural logarithm, $\ln(e)$) is also 1. This identity often appears in more complex problems and is a quick way to simplify terms.

With these properties in our toolkit, we are well-equipped to tackle a wide array of logarithmic expressions. The key is to recognize which rule applies to each situation and to apply them methodically. Now, let’s circle back to our original problem and see these properties in action.

Step-by-Step Solution

Now, let's apply these properties to our expression:

$$\log _5 \sqrt{x}-\log _5 x^9$$

Step 1: Rewrite the square root as a power

First, we recognize that $\sqrt{x}$ can be written as $x^{\frac{1}{2}}$. This gives us:

$$\log _5 x^{\frac{1}{2}}-\log _5 x^9$$

Converting the square root to a power is a fundamental step in simplifying expressions involving radicals. This conversion makes it easier to apply logarithmic properties, particularly the power rule, which we'll use in the next step. By expressing the square root as a fractional exponent, we bring it into a form that aligns perfectly with logarithmic operations, allowing for a smoother simplification process. This initial transformation is a key technique in handling expressions that mix radicals and logarithms, setting the stage for further simplification.

Step 2: Apply the power rule

Next, we'll use the power rule, which states that $\log_b(x^p) = p \log_b(x)$. Applying this rule to both terms, we get:

$$\frac{1}{2} \log _5 x - 9 \log _5 x$$

The power rule is a game-changer when it comes to dealing with exponents inside logarithms. It allows us to move the exponents out as coefficients, effectively turning a complex power within the logarithm into a simple multiplication. This transformation is incredibly useful because it reduces the complexity of the expression, making it easier to combine like terms or perform other operations. In our case, by applying the power rule, we've transformed the exponents within the logarithms into coefficients, which sets us up perfectly for the next step: combining these terms.

Step 3: Combine like terms

Now, we have two terms with the same logarithmic part ($\log_5 x$). We can combine these terms by treating $\log_5 x$ as a common factor:

$$\left(\frac{1}{2} - 9\right) \log _5 x$$

Combining like terms is a fundamental algebraic technique that simplifies expressions by grouping similar elements together. In this context, both terms have the same logarithmic component, $\log_5 x$, which means we can treat it as a common factor. This step is crucial because it condenses the expression, making it more manageable and easier to understand. By factoring out the common logarithmic term, we're left with a simple arithmetic operation involving the coefficients, which streamlines the process of reaching our final simplified form. This technique not only simplifies the expression but also highlights the underlying structure and relationships within the logarithmic terms.

Step 4: Simplify the coefficient

Let's simplify the fraction:

$$\frac{1}{2} - 9 = \frac{1}{2} - \frac{18}{2} = -\frac{17}{2}$$

So, our expression becomes:

$$-\frac{17}{2} \log _5 x$$

Simplifying the coefficient is a critical step in the simplification process, as it reduces the expression to its most basic form. Here, we're dealing with the subtraction of fractions, a common arithmetic operation. By finding a common denominator and performing the subtraction, we consolidate the numerical part of the expression into a single, simplified fraction. This not only cleans up the appearance of the expression but also makes it easier to interpret and use in further calculations. A simplified coefficient ensures that the expression is in its most concise form, ready for any subsequent mathematical operations or analyses.

Final Answer

Therefore, the expression as a single logarithm is:

$$-\frac{17}{2} \log _5 x$$

So there you have it! We've successfully expressed the given logarithmic expression as a single logarithm by applying the power rule and combining like terms. Remember, the key to mastering these simplifications is practice, so keep at it!

Common Mistakes to Avoid

When working with logarithms, there are a few common pitfalls that students often stumble into. Being aware of these can save you a lot of headaches and help ensure you arrive at the correct answer. Let’s walk through some of these common mistakes so you can steer clear of them.

One of the most frequent errors is misapplying the logarithmic properties. For instance, some students might incorrectly try to apply the quotient rule to terms that aren't in the correct form. Remember, the quotient rule applies when you have the difference of two logarithms with the same base, not when you have logarithms of differences. Similarly, the product rule applies to the logarithm of a product, not the product of logarithms. To avoid this, always double-check that the structure of your expression matches the conditions required for each property before applying it. Keeping the properties written down and referencing them as you work can be a great way to prevent these misapplications.

Another common mistake is incorrectly handling exponents and coefficients. It’s easy to get mixed up when moving exponents inside and outside of logarithms. Remember, the power rule states that $\log_b(x^p) = p \log_b(x)$. This means that an exponent on the argument inside the logarithm becomes a coefficient outside the logarithm. Students sometimes mistakenly try to apply this rule in reverse or confuse it with other logarithmic properties. To prevent this, take each step slowly and deliberately, and always double-check your work. Writing out each step and clearly indicating which property you are using can help maintain clarity and accuracy.

Errors also often arise when simplifying expressions too quickly. Logarithmic expressions can sometimes look intimidating, and there’s a temptation to rush through the simplification process. However, skipping steps or doing multiple operations at once can lead to mistakes. For example, when combining like terms, ensure you are only combining terms that have the exact same logarithmic part. Jumping ahead without careful consideration can lead to incorrect combinations and ultimately, the wrong answer. The best approach is to break the problem down into smaller, manageable steps and to check each step before moving on.

Another pitfall is forgetting the base of the logarithm. The base is a critical part of the logarithm, and it must be consistent when applying logarithmic properties. You can only combine logarithms using the product, quotient, or power rules if they have the same base. Forgetting or ignoring the base can lead to incorrect simplifications. Always make sure to explicitly write the base and verify that it is consistent across all terms before applying any properties. This is particularly important in problems where the base is not explicitly written but is implied (e.g., the common logarithm, which has a base of 10).

Lastly, students sometimes make mistakes with basic algebraic operations, such as adding fractions or simplifying coefficients. Logarithmic problems often involve fractions and negative numbers, so it’s essential to have a solid foundation in these basic skills. A mistake in arithmetic can derail the entire simplification process. If you find yourself struggling with these aspects, it may be helpful to review basic algebraic techniques and practice these skills separately. Double-checking your arithmetic and using a calculator for complex calculations can also help prevent these errors.

By being mindful of these common mistakes and taking a careful, methodical approach, you can greatly improve your accuracy and confidence when working with logarithmic expressions. Remember, practice makes perfect, so keep working on these skills, and you’ll become more adept at simplifying logarithms.

Practice Problems

To really solidify your understanding of simplifying logarithmic expressions, practice is key! Working through a variety of problems will help you become more comfortable with the properties and techniques we've discussed. Here are a few practice problems for you to tackle. Grab a pen and paper, and let’s put your skills to the test!

Problem 1: Express the following expression as a single logarithm:

$$2 \log_3(x) + \log_3(y) - \frac{1}{2} \log_3(z)$$

This problem combines several logarithmic properties, so it’s a great way to see if you can apply them in the correct order. Start by using the power rule to deal with the coefficients, then use the product and quotient rules to combine the logarithms into a single expression. Remember to pay close attention to the signs and bases to avoid errors.

Problem 2: Simplify the following expression:

$$\log_5(25x) - \log_5(x^2)$$

In this problem, you’ll need to use the quotient rule and possibly the product rule, depending on how you approach it. Don't forget to simplify any terms inside the logarithms before combining them. This exercise will help you practice identifying the correct rules to apply and simplifying expressions within logarithms.

Problem 3: Express as a single logarithm:

$$\log(100) + 3 \log(x) - \log(10)$$

This problem involves common logarithms (base 10), so remember that $\log(100)$ and $\log(10)$ can be simplified directly. Use the power rule and then combine the terms using the product and quotient rules. This exercise will reinforce your understanding of common logarithms and the application of logarithmic properties.

Problem 4: Condense the expression:

$$\frac{1}{3} [\log_2(x) + \log_2(y)] - 2 \log_2(z)$$

This problem includes a fraction outside the brackets, so be sure to distribute it correctly. Use the product rule inside the brackets first, then apply the power rule and the quotient rule to combine the terms. This problem will challenge your ability to manage multiple operations and keep track of the order of operations.

Problem 5: Write as a single logarithm:

$$4 \log_b(x) - \frac{1}{2} \log_b(y) + 3 \log_b(z)$$

This problem is similar to the first one but includes more terms, giving you additional practice with the power rule, product rule, and quotient rule. Work through it step by step, and you’ll be well on your way to mastering logarithmic simplifications.

As you work through these problems, remember to show your steps clearly and double-check your work. Logarithmic expressions can be tricky, but with practice, you'll become more confident in your ability to simplify them. If you get stuck, review the properties and examples we’ve discussed, and don’t hesitate to seek help from a teacher or tutor. Happy simplifying!

Conclusion

Alright, guys! We've journeyed through the process of expressing logarithmic expressions as a single logarithm, focusing on the expression $\log _5 \sqrt{x}-\log _5 x^9$. We broke down the problem step-by-step, highlighting the importance of the power rule and combining like terms. Remember, the properties of logarithms are your best friends when tackling these types of problems.

Consistent practice is the key to mastering these concepts. By working through various examples and practice problems, you'll build confidence and develop a solid understanding of how to manipulate logarithmic expressions. So, keep practicing, and you'll be simplifying logarithms like a pro in no time!