Solve Logarithmic Expressions Type Numerals, Not Words

Hey guys! 👋 Ever stumbled upon a math problem that looks like it's speaking another language? Well, logarithmic expressions might seem intimidating at first, but trust me, they're totally conquerable! In this article, we're going to break down a classic example step-by-step and show you how to type the correct answer in the box using numerals instead of words. Let's dive in and make these expressions our friends! 💪

Understanding the Problem

So, the problem we're tackling today is:

$$\log _2 8+\log _3\left(\frac{1}{3}\right)=\square$$

Before we even think about plugging in numbers, let's decode what this expression is actually asking. The expression includes two logarithmic terms that we need to evaluate separately and then combine. Understanding logarithmic expressions is key to solving this problem. Each term is a logarithm with a specific base and argument. It's crucial to grasp the fundamental principles of logarithms to tackle this challenge effectively.

Deciphering Logarithms

First things first, let's refresh our memory on what a logarithm actually represents. A logarithm answers the question: "To what power must we raise the base to get the argument?" Okay, let's break that down even further. The logarithm consists of three main parts:

• Base: The small number written below the log symbol (e.g., the '2' in ${\log _2 8}$).
• Argument: The number inside the parentheses (e.g., the '8' in ${\log _2 8}$).
• Value: The answer to the logarithmic expression (the exponent we're looking for).

In the expression ${\log _2 8}$, we're asking: "To what power must we raise 2 to get 8?" Similarly, for ${\log _3\left(\frac{1}{3}\right)}$, we're asking: "To what power must we raise 3 to get ${\frac{1}{3}}$?"

Breaking Down the Expression

Now that we understand the basics, let's look at our original expression again:

$$\log _2 8+\log _3\left(\frac{1}{3}\right)=\square$$

We can see that it's made up of two separate logarithmic terms connected by an addition sign. This means we'll need to evaluate each logarithm individually and then add the results together. This step-by-step approach is vital in simplifying complex logarithmic expressions. We're going to take each term, figure out its value, and then combine them to find our final answer. So, let's get started!

Solving the First Logarithm: log₂8

Alright, let's tackle the first part of our expression: ${\log _2 8}$. Remember, this is asking us: "To what power must we raise 2 to get 8?" This is a fundamental question when solving logarithmic equations, and answering it correctly is crucial. We need to find the exponent that, when applied to the base (2), yields the argument (8).

Finding the Exponent

Think about it... 2 times itself how many times gives us 8? Let's try it out:

• 2¹ = 2
• 2² = 4
• 2³ = 8

Bingo! 2 raised to the power of 3 equals 8. So, the value of ${\log _2 8}$ is 3. We've successfully evaluated the first term by carefully considering the powers of the base. Understanding the relationship between exponents and logarithms is key to solving these types of problems.

Writing the Solution

Now that we've figured out the answer, let's write it down: ${\log _2 8 = 3}$. Great job! We've conquered the first part of our expression. This step demonstrates how to evaluate basic logarithms and sets the stage for tackling more complex problems. By breaking down the problem into smaller parts, we make the solution much more accessible.

Solving the Second Logarithm: log₃(1/3)

Okay, time to move on to the second part of our expression: ${\log _3\left(\frac{1}{3}\right)}$. This one might look a little trickier because of the fraction, but don't worry, we've got this! Just like before, we need to figure out: "To what power must we raise 3 to get ${\frac{1}{3}}$?" This problem highlights the importance of understanding negative exponents in logarithms.

Dealing with Fractions and Negative Exponents

Remember, a negative exponent means we're dealing with a reciprocal. In other words, ${x^{-n} = \frac{1}{x^n}}$. So, we're looking for a power that, when applied to 3, will give us its reciprocal, ${\frac{1}{3}}$. The ability to relate fractions to negative exponents is a crucial skill in mathematics.

Finding the Exponent

Let's think about it. We know that 3 to the power of 1 is just 3. To get ${\frac{1}{3}}$, we need to use a negative exponent. Specifically, 3 to the power of -1 is equal to ${\frac{1}{3}}$. This is because 3⁻¹ = ${\frac{1}{3^1}}$ = ${\frac{1}{3}}$. So, the value of ${\log _3\left(\frac{1}{3}\right)}$ is -1. This demonstrates how logarithms handle fractional arguments using negative exponents.

Writing the Solution

Awesome! We've solved the second logarithm. We can write it as: ${\log _3\left(\frac{1}{3}\right) = -1}$. Now, we're one step closer to the final answer. Understanding the role of negative exponents in logarithmic equations is essential for accurate problem-solving.

Combining the Results

We've done the hard work! We've figured out that ${\log _2 8 = 3}$ and ${\log _3\left(\frac{1}{3}\right) = -1}$. Now, all that's left to do is combine these results according to the original expression. This step emphasizes the importance of order of operations in mathematical expressions.

Adding the Values

Our original expression was:

$$\log _2 8+\log _3\left(\frac{1}{3}\right)=\square$$

We know that ${\log _2 8 = 3}$ and ${\log _3\left(\frac{1}{3}\right) = -1}$. So, we can substitute these values into the expression:

$$3 + (-1) = \square$$

Now it's simple addition! 3 plus -1 equals 2. Therefore, the value of the expression is 2. This step highlights the additive property of combined logarithmic expressions.

The Final Answer

We did it! 🎉 We've successfully evaluated the logarithmic expression. The final answer is 2. This process demonstrates how to solve logarithmic expressions step-by-step, leading to the correct solution.

Typing the Correct Answer

Okay, now for the final step: typing the correct answer in the box using numerals instead of words. We've already calculated that the value of the expression is 2. So, all you need to do is type "2" in the box. Remember, the question specifically asks for numerals, not words, so writing "two" would be incorrect. This emphasizes the importance of following instructions carefully in mathematical problems.

Double-Checking Your Work

Before submitting your answer, it's always a good idea to double-check your work. Make sure you've correctly evaluated each logarithm and performed the addition accurately. This practice helps to minimize errors in mathematical calculations. A quick review can save you from making simple mistakes and ensure you get the correct answer.

Conclusion: Mastering Logarithmic Expressions

And there you have it! We've successfully solved a logarithmic expression and learned how to type the correct answer in the box using numerals. We started by understanding the basics of logarithms, then broke down the problem step-by-step, and finally combined the results to find our solution. Remember, the key to mastering logarithmic expressions is practice and a solid understanding of the fundamentals. Understanding the principles behind logarithmic functions and their applications is crucial for success in mathematics. Keep practicing, and you'll become a logarithm pro in no time! 💪

SEO Title: Solve Logarithmic Expressions: Type Numerals, Not Words

Repair Input Keyword: What is the value of the expression log₂8 + log₃(1/3)?