Solving 3^(2x) + 12 = 39 A Step-by-Step Guide

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Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of exponential equations. We're going to tackle a specific problem: What is the solution to the equation 3^(2x) + 12 = 39? Stick around as we break down the steps, explore the concepts, and arrive at the correct answer. We'll make sure to clarify every stage, so you'll be solving similar equations like a pro in no time. Let's get started and demystify this mathematical puzzle together!

Understanding Exponential Equations

Before we jump into solving our equation, let's briefly discuss what exponential equations are and why they're so important. Exponential equations are equations in which the variable appears in the exponent. They pop up in various real-world applications, from calculating compound interest and population growth to modeling radioactive decay and the spread of diseases. Understanding how to solve them is, therefore, a crucial skill in many fields.

These equations have a general form, often expressed as a^x = b, where 'a' is the base, 'x' is the exponent, and 'b' is the result. The key to solving exponential equations lies in manipulating them to isolate the variable in the exponent. We often use logarithms to achieve this, but in simpler cases, we can also use algebraic techniques like the one we'll employ today. The power of exponential equations comes from their ability to describe situations where quantities grow or decay at a rate proportional to their current value. This makes them incredibly useful for predicting future outcomes based on current trends.

For example, in finance, exponential equations help us understand how investments grow over time, thanks to the magic of compound interest. In biology, they model how populations can explode or decline depending on factors like birth rates and mortality rates. In physics, they are essential for describing radioactive decay, which is the process by which unstable atomic nuclei lose energy by emitting radiation. So, whether you're crunching numbers for a business plan, studying the dynamics of an ecosystem, or exploring the fundamental laws of the universe, understanding exponential equations is a valuable asset.

Now, with a solid grasp of what exponential equations are all about, let's roll up our sleeves and solve the equation at hand: 3^(2x) + 12 = 39. We'll take it step by step, making sure you're comfortable with each move we make. Remember, the goal is to isolate 'x' and find its value. So, let’s get to it and unlock the solution!

Step-by-Step Solution to 3^(2x) + 12 = 39

Alright, let's get our hands dirty and solve this equation step-by-step. Our mission, should we choose to accept it, is to find the value of 'x' that satisfies the equation 3^(2x) + 12 = 39. Don't worry, it's not as daunting as it might look! We'll break it down into manageable chunks.

Step 1: Isolate the Exponential Term

First things first, we want to get the exponential term, which is 3^(2x), all by itself on one side of the equation. To do this, we need to get rid of that '+ 12'. The obvious move here is to subtract 12 from both sides of the equation. This keeps the equation balanced and helps us isolate the term we're interested in.

So, we have:

3^(2x) + 12 - 12 = 39 - 12

This simplifies beautifully to:

3^(2x) = 27

Great! We've made significant progress. Now our equation looks much cleaner and easier to handle. The exponential term is nicely isolated, and we can move on to the next step.

Step 2: Express Both Sides with the Same Base

Now, this is where things get a little clever. To solve for 'x', we need to find a way to compare the exponents. A fantastic strategy for this is to express both sides of the equation using the same base. Looking at our equation, 3^(2x) = 27, we can see that 27 can also be written as a power of 3. In fact, 27 is 3 cubed (3^3). So, let's rewrite 27 as 3^3.

Our equation now becomes:

3^(2x) = 3^3

See how we did that? By expressing both sides with the same base (which is 3 in this case), we've set ourselves up for the next crucial step. This is a common technique when dealing with exponential equations, so it’s worth remembering.

Step 3: Equate the Exponents

This is the exciting part! Now that we have the same base on both sides of the equation, we can directly equate the exponents. If 3 raised to some power is equal to 3 raised to another power, then those powers must be equal. It's a fundamental property of exponential functions that makes solving these equations possible.

So, since 3^(2x) = 3^3, we can confidently say that:

2x = 3

Boom! We've transformed our exponential equation into a simple linear equation. The variable 'x' is no longer trapped in the exponent. We're almost home now.

Step 4: Solve for x

Okay, this is the final stretch. We have the equation 2x = 3, and we need to solve for 'x'. This is a piece of cake. To isolate 'x', we simply divide both sides of the equation by 2.

So, we get:

x = 3/2

And there you have it! We've found the value of 'x' that satisfies the original equation. It's like cracking a secret code, isn't it?

Therefore, the solution to the equation 3^(2x) + 12 = 39 is x = 3/2. It was a fun journey, and we conquered it together! Now, let's take a look at the answer choices and see which one matches our solution.

Identifying the Correct Answer Choice

Fantastic work, team! We've meticulously worked through the equation 3^(2x) + 12 = 39 and arrived at the solution x = 3/2. Now, let's circle back to the answer choices provided and pinpoint the one that matches our result.

The answer choices given were:

A. x = 9/2 B. x = 3 C. x = 9 D. x = 3/2

A quick glance at our solution, x = 3/2, reveals that it perfectly aligns with option D. So, the correct answer is indeed D. x = 3/2.

It's always a satisfying moment when our hard work pays off and we can confidently identify the correct answer. By breaking down the problem into smaller, manageable steps, we were able to navigate the equation with clarity and precision. This approach is key to tackling any mathematical challenge, so keep practicing and building your skills!

Now that we've nailed the solution, let's take a moment to reflect on the process we used. Understanding the steps involved is just as important as getting the right answer, as it equips us to solve similar problems in the future. So, let's recap the key moves we made along the way.

Recapping the Solution Process

Alright, mathletes, let's take a moment to zoom out and recap the journey we took to solve the equation 3^(2x) + 12 = 39. Understanding the process is just as crucial as getting the correct answer because it equips us with the tools to tackle similar problems down the road. So, let's break it down:

  1. Isolate the Exponential Term: Our first move was to isolate the exponential term, 3^(2x). We did this by subtracting 12 from both sides of the equation. This gave us a cleaner equation to work with: 3^(2x) = 27. Isolating the term is often the crucial first step in solving various types of equations.

  2. Express Both Sides with the Same Base: Next up, we got a little clever and rewrote both sides of the equation using the same base. We recognized that 27 could be expressed as 3^3. This transformation allowed us to compare the exponents directly: 3^(2x) = 3^3. This step highlights the importance of recognizing patterns and using mathematical properties to simplify problems.

  3. Equate the Exponents: With the same base on both sides, we were able to equate the exponents. This is a fundamental principle of exponential equations: if a^m = a^n, then m = n. So, we deduced that 2x = 3. This step demonstrates the power of understanding the underlying principles of mathematical operations.

  4. Solve for x: Finally, we solved the resulting linear equation for x. Dividing both sides of 2x = 3 by 2, we found that x = 3/2. This final step showcases the importance of basic algebraic skills in solving more complex problems.

By following these steps, we transformed a seemingly complex exponential equation into a straightforward solution. Remember, the key to mastering math is to break down problems into smaller, manageable parts and to understand the underlying principles. With practice and a clear understanding of the process, you'll be solving exponential equations like a pro in no time!

Now, let's broaden our horizons a bit and explore other methods that we could have used to solve this equation. While our approach was efficient, there's often more than one way to skin a mathematical cat, so to speak. Let's dive into some alternative approaches.

Alternative Methods for Solving Exponential Equations

Okay, folks, let's switch gears for a moment and explore some alternative routes to solving exponential equations. While we successfully tackled our problem using a specific method, it's always beneficial to have a few tricks up our sleeves. Different problems might lend themselves better to different approaches, and having a versatile toolkit can make you a true math whiz.

1. Using Logarithms

One of the most powerful tools for solving exponential equations is logarithms. Remember that logarithms are the inverse operation of exponentiation. If we have an equation like a^x = b, we can take the logarithm of both sides to bring the exponent down.

In our case, we had 3^(2x) = 27. We could take the logarithm base 3 of both sides:

log₃(3^(2x)) = log₃(27)

Using the property of logarithms that logₐ(a^b) = b, we get:

2x = log₃(27)

Since 27 = 3^3, log₃(27) = 3, so:

2x = 3

And we're back to the same linear equation we had before, which we can easily solve for x = 3/2.

This method is particularly useful when you can't easily express both sides of the equation with the same base. Logarithms provide a universal tool for unwrapping the exponent.

2. Graphical Methods

Another way to approach exponential equations is graphically. We can think of the equation 3^(2x) + 12 = 39 as finding the intersection point of two functions:

  • y = 3^(2x) + 12
  • y = 39

If we were to graph these two functions, the x-coordinate of their intersection point would be the solution to our equation. This method is great for visualizing the solution and can be particularly helpful for equations that are difficult to solve algebraically. While it might not give you an exact answer without the aid of technology, it can provide a good estimate and a deeper understanding of the equation's behavior.

3. Numerical Methods

For equations that are really tough to crack algebraically, we can turn to numerical methods. These methods involve using iterative processes to approximate the solution. One common approach is the Newton-Raphson method, which uses calculus to refine an initial guess until it converges on the true solution. Numerical methods are often used in computer programs and calculators to solve complex equations.

So, there you have it – a few alternative ways to tackle exponential equations. Each method has its strengths and weaknesses, and the best approach often depends on the specific problem at hand. By understanding these different techniques, you'll be well-equipped to handle a wide range of mathematical challenges.

Real-World Applications of Exponential Equations

Okay, guys, we've nailed the solution to our equation and explored some alternative methods. But let's take a step back and think about the bigger picture. Why are we even learning about exponential equations in the first place? The answer, my friends, is that they pop up all over the place in the real world!

Exponential equations are like the Swiss Army knives of mathematics – incredibly versatile and useful in a wide array of fields. From finance to biology to physics, these equations help us model and understand phenomena that grow or decay at a rate proportional to their current value. Let's explore some specific examples.

1. Compound Interest

One of the most common applications of exponential equations is in finance, specifically when calculating compound interest. When you invest money and earn interest, that interest can then earn more interest, and so on. This compounding effect leads to exponential growth.

The formula for compound interest is:

A = P (1 + r/n)^(nt)

Where:

  • A is the final amount
  • P is the principal (initial investment)
  • r is the annual interest rate
  • n is the number of times interest is compounded per year
  • t is the number of years

See that exponent there? That's what makes this an exponential equation. It allows us to predict how investments will grow over time, and it's a crucial tool for financial planning.

2. Population Growth

Exponential equations also play a vital role in modeling population growth. In ideal conditions, populations tend to grow exponentially, meaning their growth rate is proportional to their size. This is because the more individuals there are, the more offspring they can produce.

The basic formula for exponential population growth is:

N(t) = N₀ * e^(rt)

Where:

  • N(t) is the population at time t
  • N₀ is the initial population
  • e is the base of the natural logarithm (approximately 2.71828)
  • r is the growth rate
  • t is time

This equation helps us understand how populations can explode or decline under different conditions. It's used by ecologists, demographers, and policymakers to study and manage populations.

3. Radioactive Decay

In the realm of physics, exponential equations are essential for describing radioactive decay. Radioactive substances decay over time, meaning their atoms spontaneously transform into other atoms. The rate of this decay is exponential, with the amount of substance decreasing by a constant fraction per unit of time.

The formula for radioactive decay is:

N(t) = N₀ * e^(-λt)

Where:

  • N(t) is the amount of substance remaining at time t
  • N₀ is the initial amount of substance
  • e is the base of the natural logarithm
  • λ (lambda) is the decay constant
  • t is time

This equation is used in carbon dating, nuclear medicine, and various other applications where radioactive materials are involved.

These are just a few examples of the many real-world applications of exponential equations. They are powerful tools for understanding and predicting phenomena in finance, biology, physics, and beyond. So, the next time you encounter an exponential equation, remember that you're not just solving an abstract problem – you're unlocking a piece of the puzzle of how the world works!

Conclusion: Mastering Exponential Equations

Wow, what a journey we've been on! We started with a seemingly simple equation, 3^(2x) + 12 = 39, and we've not only solved it but also delved into the fascinating world of exponential equations. We've explored their fundamental principles, alternative solution methods, and their widespread applications in the real world.

By breaking down the problem step-by-step, we've shown how to isolate the exponential term, express both sides with the same base, equate the exponents, and solve for the variable. We've also seen how logarithms, graphical methods, and numerical methods can be used to tackle exponential equations from different angles. And we've discovered how these equations are essential tools for understanding compound interest, population growth, radioactive decay, and much more.

The key takeaway here is that mastering exponential equations isn't just about memorizing formulas or following procedures. It's about understanding the underlying concepts and developing a problem-solving mindset. It's about recognizing patterns, applying mathematical properties, and thinking critically about the steps involved.

So, what's the best way to become a master of exponential equations? Practice, practice, practice! The more you work with these equations, the more comfortable and confident you'll become. Try solving different types of problems, exploring alternative methods, and seeking out real-world examples to deepen your understanding.

Remember, mathematics is not a spectator sport. It's something you learn by doing. So, roll up your sleeves, grab a pencil and paper, and dive into the world of exponential equations. You might be surprised at how much you enjoy the challenge and how powerful these tools can be.

Keep exploring, keep learning, and keep pushing your mathematical boundaries. The world of exponential equations – and mathematics in general – is vast and full of exciting discoveries waiting to be made. So, go out there and make some mathematical magic happen!