Solving Logarithmic Equations: Step-by-Step Guide

Hey guys! Let's dive into the world of logarithmic equations and figure out how to solve them. Today, we're tackling a specific question: What's the very first step in solving the equation log₂(x) + log₂(x - 6) = 4? If you've ever felt lost staring at logs, don't worry, we'll break it down together. We'll explore the fundamental properties of logarithms, apply them to this equation, and nail down that crucial first step. So, grab your thinking caps, and let's get started!

Understanding Logarithms

Before we jump into solving our equation, let's make sure we're all on the same page about what logarithms actually are. At its heart, a logarithm is simply the inverse of an exponential function. Think of it this way: if 2³ = 8, then log₂(8) = 3. The logarithm (base 2 in this case) tells you what power you need to raise the base to in order to get a specific number. Understanding this relationship between logarithms and exponents is absolutely key to manipulating logarithmic equations.

• The Logarithmic Function: The general form of a logarithmic function is logₐ(x) = y, which is equivalent to aʸ = x. Here, 'a' is the base (a positive number not equal to 1), 'x' is the argument (the number we're taking the logarithm of), and 'y' is the exponent.
• Common Logarithms: When you see 'log' without a base written (like log(x)), it usually means the base is 10. This is called the common logarithm.
• Natural Logarithms: Another important logarithm is the natural logarithm, written as ln(x), where the base is the mathematical constant 'e' (approximately 2.71828).

Now, why are logarithms so important? Well, they show up in all sorts of fields, from science and engineering to finance and computer science. They're particularly useful for dealing with very large or very small numbers, as they allow us to compress scales and make calculations more manageable. For instance, the Richter scale for measuring earthquake intensity is logarithmic, as is the decibel scale for measuring sound levels. Mastering logarithms opens up a whole new world of problem-solving possibilities!

Key Properties of Logarithms

To solve logarithmic equations effectively, you absolutely need to know the key properties of logarithms. These properties allow us to manipulate and simplify expressions, which is crucial for isolating the variable we're trying to solve for. Here are the most important ones:

1. Product Rule: logₐ(mn) = logₐ(m) + logₐ(n). This rule tells us that the logarithm of a product is equal to the sum of the logarithms of the individual factors.
2. Quotient Rule: logₐ(m/n) = logₐ(m) - logₐ(n). The logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator.
3. Power Rule: logₐ(mᵖ) = p * logₐ(m). The logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number.
4. Change of Base Rule: log♭(a) = logₓ(a) / logₓ(b). This allows you to convert logarithms from one base to another, which is particularly useful when using calculators that only have common or natural logarithm functions.
5. Logarithm of 1: logₐ(1) = 0. Any number raised to the power of 0 equals 1.
6. Logarithm of the Base: logₐ(a) = 1. Any number raised to the power of 1 equals itself.

These properties are our secret weapons when it comes to solving logarithmic equations. By understanding and applying them correctly, we can transform complex equations into simpler, more manageable forms. So, keep these rules handy as we move on to tackling our specific problem.

Identifying the First Step

Okay, let's get back to the equation we're trying to solve: log₂(x) + log₂(x - 6) = 4. The big question is, what's the very first step we should take? If we look at the answer choices provided, we can see that they all involve some kind of manipulation of the logarithmic terms on the left side of the equation. This gives us a big clue that our initial focus should be on simplifying that side using the properties of logarithms we just discussed.

Now, let's think about which property might be most helpful here. We have the sum of two logarithms with the same base (base 2). Does that ring a bell? If you remember the product rule, you're on the right track! The product rule states that logₐ(mn) = logₐ(m) + logₐ(n). In other words, the sum of two logarithms with the same base is equal to the logarithm of the product of their arguments.

So, how does this apply to our equation? Well, we can think of log₂(x) as logₐ(m) and log₂(x - 6) as logₐ(n). Applying the product rule, we can combine these two logarithmic terms into a single logarithm: log₂[x(x - 6)]. This means we're multiplying the arguments 'x' and '(x - 6)' together inside the logarithm.

Therefore, the first step in solving the equation log₂(x) + log₂(x - 6) = 4 is to use the product rule to rewrite the left side as log₂[x(x - 6)] = 4. This single step significantly simplifies the equation and gets us closer to isolating the variable 'x'. Now, let's take a look at the answer choices and see which one matches this first step.

Analyzing the Answer Choices

We've determined that the first step in solving the equation log₂(x) + log₂(x - 6) = 4 is to apply the product rule and rewrite the left side as log₂[x(x - 6)]. Now, let's carefully examine the answer choices to see which one matches this transformation:

A. log₂[x(x - 6)] = 4 B. x + x - 6 = 4 C. x + x - 6 = 2⁴ D. log₂ (x / (x - 6)) = 4

Looking at these options, it's pretty clear that option A, log₂[x(x - 6)] = 4, is the correct first step. This directly applies the product rule, combining the two logarithmic terms on the left side of the equation into a single logarithm. The other options either skip this crucial step or apply the wrong logarithmic property.

• Option B, x + x - 6 = 4, seems to completely ignore the logarithms and incorrectly adds the arguments. This is a common mistake for those not fully grasping the properties of logarithms.
• Option C, x + x - 6 = 2⁴, takes a step towards exponentiating but misses the crucial initial step of combining the logarithms. It's jumping ahead without properly simplifying the equation first.
• Option D, log₂ (x / (x - 6)) = 4, uses the quotient rule instead of the product rule. This would be appropriate if we had a subtraction of logarithms, not an addition.

So, we can confidently say that option A is the correct first step. It demonstrates a clear understanding of the product rule and its application in simplifying logarithmic equations. But what about the subsequent steps? Let's briefly discuss how we would continue solving the equation to fully understand the process.

Continuing the Solution

Now that we've nailed down the first step, let's think about what comes next. We've transformed the equation into log₂[x(x - 6)] = 4. The goal now is to isolate 'x', but it's currently trapped inside the logarithm. How do we get it out?

The key here is to remember the relationship between logarithms and exponents. Since log₂(y) = z is equivalent to 2ᶻ = y, we can rewrite our equation in exponential form. In this case, the base is 2, the exponent is 4, and the argument is x(x - 6). So, we can rewrite the equation as 2⁴ = x(x - 6).

This simplifies to 16 = x(x - 6). Now we have a quadratic equation, which we can solve using standard algebraic techniques:

1. Expand the right side: 16 = x² - 6x
2. Move all terms to one side: 0 = x² - 6x - 16
3. Factor the quadratic: 0 = (x - 8)(x + 2)
4. Solve for x: x = 8 or x = -2

However, we're not quite done yet! It's crucial to check our solutions in the original equation. Logarithms are only defined for positive arguments. If we plug x = -2 back into the original equation, we get log₂(-2), which is undefined. Therefore, x = -2 is an extraneous solution and must be discarded.

If we plug x = 8 back into the original equation, we get log₂(8) + log₂(8 - 6) = log₂(8) + log₂(2) = 3 + 1 = 4, which is correct. So, the only valid solution is x = 8.

This illustrates the importance of not only knowing the properties of logarithms but also understanding the domain restrictions and checking for extraneous solutions. Solving logarithmic equations requires a careful and methodical approach.

Conclusion

So, guys, we've successfully navigated the world of logarithmic equations and pinpointed the first crucial step in solving log₂(x) + log₂(x - 6) = 4. We learned that applying the product rule, which combines the sum of logarithms into a single logarithm, is the key to unlocking this problem. We also saw how understanding the fundamental properties of logarithms and the relationship between logarithms and exponents is essential for tackling these types of equations.

Remember, solving logarithmic equations is like building a house – you need a solid foundation. Knowing the properties, understanding the domain restrictions, and checking for extraneous solutions are all part of that foundation. With practice and a clear understanding of these concepts, you'll be solving logarithmic equations like a pro in no time! Keep practicing, and don't be afraid to ask questions. You've got this!