Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of logarithms to solve a specific equation: log₂(9x) - log₂3 = 3. If you've ever felt a bit puzzled by logs, don't worry, we'll break it down step by step in a way that's easy to understand. So, grab your thinking caps, and let's get started!
Understanding Logarithms: The Basics
Before we jump into the solution, let's quickly refresh our understanding of logarithms. At its heart, a logarithm answers the question: "To what power must we raise a base to get a certain number?" For instance, log₂8 asks, "To what power must we raise 2 to get 8?" The answer, of course, is 3 because 2³ = 8. So, log₂8 = 3. Understanding this fundamental concept is crucial for tackling logarithmic equations.
Logarithms are essentially the inverse operation of exponentiation. They help us solve for exponents in equations. The expression logₐb = c is equivalent to saying aᶜ = b. Here, 'a' is the base of the logarithm, 'b' is the argument (the number we're taking the logarithm of), and 'c' is the exponent or the result of the logarithm. In our equation, log₂(9x) - log₂3 = 3, we have a base of 2, and we're dealing with expressions involving 'x' inside the logarithms. Our goal is to isolate 'x' and find its value. Remember, the key to solving logarithmic equations often lies in using the properties of logarithms to simplify the equation and then converting it into exponential form.
Deciphering the Equation: log₂(9x) - log₂3 = 3
Now, let's dissect the equation we're tackling: log₂(9x) - log₂3 = 3. Notice that we have two logarithmic terms on the left side, both with the same base (2). This is excellent news because it means we can use one of the fundamental properties of logarithms to simplify things. This property states that the logarithm of a quotient is equal to the difference of the logarithms. In mathematical terms, logₐ(b/c) = logₐb - logₐc. This property is a game-changer when dealing with subtraction between logarithmic terms, as it allows us to combine them into a single logarithm.
Applying this property to our equation, we can rewrite log₂(9x) - log₂3 as log₂(9x/3). This simplification is a crucial step because it reduces the two logarithmic terms into one, making the equation much easier to handle. So, our equation now looks like this: log₂(9x/3) = 3. Before we move on, let's simplify the argument inside the logarithm further. We have 9x divided by 3, which simplifies to 3x. Thus, our equation transforms into log₂(3x) = 3. We've successfully condensed the original equation into a more manageable form, and we're one step closer to finding the value of 'x'.
Applying Logarithmic Properties: Condensing the Expression
Before we dive into solving for 'x', let's solidify our understanding of the logarithmic property we used. The property logₐb - logₐc = logₐ(b/c) is a cornerstone of logarithmic manipulation. It allows us to combine logarithmic terms with the same base when they are being subtracted. This is particularly useful in equations like ours, where multiple logarithmic terms can obscure the solution. By condensing them into a single term, we simplify the equation and make it easier to isolate the variable.
The reverse of this property, logₐ(b/c) = logₐb - logₐc, is equally important. It allows us to expand a single logarithmic term into multiple terms, which can be helpful in other scenarios. Understanding both directions of this property gives us flexibility in manipulating logarithmic expressions. Remember, the key is to recognize when applying this property will simplify the equation and move us closer to the solution. In our case, condensing the two logarithmic terms was a pivotal step in simplifying log₂(9x) - log₂3 = 3 into log₂(3x) = 3. Now, with our simplified equation, we're ready to move on to the next stage: converting the logarithmic equation into its exponential form.
Converting to Exponential Form: Unveiling the Solution
With our equation simplified to log₂(3x) = 3, the next step is to convert it from logarithmic form to exponential form. This conversion is the key to unlocking the value of 'x'. Remember the fundamental relationship between logarithms and exponents: logₐb = c is equivalent to aᶜ = b. Applying this to our equation, where the base 'a' is 2, the argument 'b' is 3x, and the result 'c' is 3, we can rewrite log₂(3x) = 3 as 2³ = 3x.
Now, we have a simple exponential equation: 2³ = 3x. We know that 2³ equals 2 * 2 * 2, which is 8. So, our equation becomes 8 = 3x. We've successfully transformed the logarithmic equation into a linear equation, which is much easier to solve. To isolate 'x', we simply need to divide both sides of the equation by 3. This gives us x = 8/3. And there you have it! We've found the solution to the equation. The value of 'x' that satisfies the original equation log₂(9x) - log₂3 = 3 is 8/3.
Isolating 'x': The Final Step
Let's recap the final steps we took to isolate 'x'. After converting the logarithmic equation log₂(3x) = 3 into its exponential form 2³ = 3x, we simplified 2³ to 8, giving us 8 = 3x. To get 'x' by itself, we performed a simple algebraic manipulation: dividing both sides of the equation by 3. This is a fundamental principle in solving equations – whatever operation you perform on one side, you must perform on the other to maintain the equality.
Dividing both sides of 8 = 3x by 3 gives us 8/3 = x. This is the final step in solving for 'x'. It's crucial to remember that the goal in solving any equation is to isolate the variable we're trying to find. In this case, 'x' was being multiplied by 3, so the inverse operation, division by 3, was the key to isolating it. We've now confidently arrived at the solution: x = 8/3. This value satisfies the original equation, and we've successfully navigated the logarithmic equation to find it.
Verifying the Solution: Ensuring Accuracy
Before we declare victory, it's always a good practice to verify our solution. This step ensures that we haven't made any errors along the way and that our answer is indeed correct. To verify our solution, we'll substitute x = 8/3 back into the original equation: log₂(9x) - log₂3 = 3. Replacing 'x' with 8/3, we get log₂(9 * 8/3) - log₂3 = 3.
Let's simplify the expression inside the first logarithm. 9 multiplied by 8/3 is (9/3) * 8, which simplifies to 3 * 8 = 24. So, our equation now looks like this: log₂24 - log₂3 = 3. Now, we can use the logarithmic property we discussed earlier: logₐb - logₐc = logₐ(b/c). Applying this property, we get log₂(24/3) = 3. Simplifying the fraction inside the logarithm, 24/3 equals 8. So, our equation becomes log₂8 = 3. Now, we ask ourselves: "To what power must we raise 2 to get 8?" The answer is 3, since 2³ = 8. Therefore, log₂8 = 3 is a true statement. Since our substitution resulted in a true statement, we can confidently conclude that our solution, x = 8/3, is correct. Verifying the solution is a crucial step in problem-solving, as it gives us assurance that our answer is accurate and that we've correctly applied the mathematical principles.
Conclusion: Mastering Logarithmic Equations
Wow, guys, we've really journeyed through the world of logarithms today! We successfully solved the equation log₂(9x) - log₂3 = 3, and along the way, we've reinforced some key concepts about logarithms and their properties. Remember, the key to solving logarithmic equations is to understand the relationship between logarithms and exponents, and to utilize the properties of logarithms to simplify the equations. We condensed logarithmic terms using the property logₐb - logₐc = logₐ(b/c), converted the equation to exponential form, and then solved for 'x'. And finally, we verified our solution to ensure its accuracy.
The solution to the equation log₂(9x) - log₂3 = 3 is indeed x = 8/3. This corresponds to option B in the choices provided. By understanding the fundamental principles and applying them step by step, you can tackle even the most challenging logarithmic equations. So, keep practicing, keep exploring, and you'll become a logarithm pro in no time! Remember, math is a journey, and every equation solved is a step forward.
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