Solving $2ln(7x) = 4$: Find The Exact Solution

by Andrew McMorgan 47 views

Hey Plastik Magazine readers! Today, we're diving into a fun math problem involving logarithms. We'll break down how to solve the equation 2extln(7x)=42 ext{ln}(7x) = 4 step-by-step. So, grab your calculators (though we won't need them for the exact solution!), and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we understand what the equation is asking. The equation 2extln(7x)=42 ext{ln}(7x) = 4 involves a natural logarithm, denoted by "ln". Remember that the natural logarithm is just a logarithm with base e, where e is Euler's number (approximately 2.71828). Essentially, we're trying to find the value of x that makes the equation true.

Logarithms are the inverse operation to exponentiation. So, if we have $ ext{ln}(a) = b$, that means eb=ae^b = a. This relationship is key to solving our equation. Also, remember that the argument of a logarithm (the thing inside the parentheses) must be positive. In our case, 7x7x must be greater than 0, which means xx must be greater than 0. This is important to keep in mind when we get our solution, as we need to make sure it's valid.

Think of it like this: the logarithm asks, "To what power must we raise the base (e in this case) to get this number?" Our equation adds a couple of extra steps with the multiplication by 2 and the multiplication of x by 7 inside the logarithm, but we'll tackle those one at a time. We're aiming to isolate x by undoing each operation in reverse order. This involves using the properties of logarithms and exponents to simplify the equation until we have x by itself on one side.

Step-by-Step Solution

Okay, let's get down to the nitty-gritty and solve this equation! We'll go through each step in detail so you can follow along easily.

Step 1: Isolate the Logarithm

Our first goal is to get the logarithmic term, $ ext{ln}(7x)$, by itself on one side of the equation. Currently, the logarithm is being multiplied by 2. To undo this, we'll divide both sides of the equation by 2:

2extln(7x)=42 ext{ln}(7x) = 4

Divide both sides by 2:

$ ext{ln}(7x) = 2$

Great! We've successfully isolated the logarithm. Now, we're one step closer to getting x alone.

Step 2: Convert to Exponential Form

Now comes the crucial step where we use the relationship between logarithms and exponents. Remember that $ ext{ln}(a) = b$ is equivalent to eb=ae^b = a. In our case, we have $ ext{ln}(7x) = 2$. So, we can rewrite this in exponential form as:

e2=7xe^2 = 7x

This step is super important because it gets rid of the logarithm and allows us to work with a simple algebraic equation.

Step 3: Isolate x

We're almost there! Now we just need to isolate x. Currently, x is being multiplied by 7. To undo this, we'll divide both sides of the equation by 7:

e2=7xe^2 = 7x

Divide both sides by 7:

x = rac{e^2}{7}

And that's it! We've found the exact solution for x.

The Exact Solution

The exact solution to the equation 2extln(7x)=42 ext{ln}(7x) = 4 is:

x = rac{e^2}{7}

Notice that we've left the answer in terms of e. This is what makes it the exact solution. If we were to plug this into a calculator, we'd get a decimal approximation, but the fraction rac{e^2}{7} is the most precise representation of the solution.

Checking the Solution

It's always a good idea to check your solution to make sure it's correct and that it makes sense in the context of the original problem. To do this, we'll substitute our solution, x = rac{e^2}{7}, back into the original equation:

2extln(7x)=42 ext{ln}(7x) = 4

Substitute x = rac{e^2}{7}:

2 ext{ln}igg(7 imes rac{e^2}{7}igg) = 4

Simplify:

2extln(e2)=42 ext{ln}(e^2) = 4

Remember that $ ext{ln}(e^2)$ is just 2 (because e raised to the power of 2 is e squared!). So we have:

2imes2=42 imes 2 = 4

4=44 = 4

Our solution checks out! This confirms that x = rac{e^2}{7} is indeed the correct solution. We also need to make sure that our solution is valid in the domain of the logarithm. Since x = rac{e^2}{7} is a positive number, it satisfies the condition that 7x>07x > 0. So, we're good to go!

Why Exact Solutions Matter

You might be wondering, "Why bother with the exact solution when I can just use a calculator to get a decimal?" Well, in many situations, especially in higher-level mathematics and scientific applications, exact solutions are crucial. Here's why:

  • Precision: Decimal approximations can introduce rounding errors. If you're using the solution in further calculations, these errors can accumulate and lead to inaccurate results. Exact solutions, on the other hand, maintain the full precision of the answer.
  • Understanding: Leaving the answer in terms of e or other mathematical constants helps us understand the structure of the solution and its relationship to the original equation. It gives us a deeper insight than just a string of digits.
  • Symbolic Manipulation: Exact solutions allow us to manipulate expressions algebraically. For example, we can simplify expressions, cancel terms, and perform other operations more easily with exact values than with decimal approximations.

So, while calculators are handy tools, it's important to appreciate the power and elegance of exact solutions.

Practice Makes Perfect

Now that we've walked through this problem together, the best way to solidify your understanding is to practice! Try solving similar equations involving logarithms and exponentials. You can find plenty of examples online or in textbooks. Remember to focus on the steps we discussed: isolating the logarithm, converting to exponential form, and isolating x. And don't forget to check your solutions!

Solving logarithmic equations can seem tricky at first, but with a little practice, you'll get the hang of it. Remember the key relationships between logarithms and exponentials, and you'll be well on your way to mastering these types of problems. Keep practicing, and you'll become a math whiz in no time!

Conclusion

So, there you have it! We've successfully solved the equation 2extln(7x)=42 ext{ln}(7x) = 4 and found the exact solution: x = rac{e^2}{7}. We've also discussed why exact solutions are important and how to check your answers. I hope this breakdown was helpful and that you feel more confident tackling similar problems in the future. Keep exploring the fascinating world of math, and remember to have fun while you're at it!