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

Hey there, math enthusiasts! Ever stumbled upon an equation that looks like a puzzle waiting to be solved? Well, today we're diving deep into one such equation: 3^(2x) + 12 = 39. This isn't just about finding the right answer; it's about understanding the process, the why behind each step. So, buckle up, because we're about to embark on a mathematical journey that will not only equip you with the solution but also enhance your problem-solving skills.

Cracking the Code: A Step-by-Step Solution

Before we jump into the nitty-gritty, let’s take a moment to appreciate the beauty of exponential equations. They might seem daunting at first, but with a systematic approach, they're totally conquerable. Our main goal here is to isolate the variable x. This involves peeling away the layers surrounding x, one operation at a time. Think of it like unwrapping a gift – each step reveals something new until we get to the treasure inside.

Step 1: Isolating the Exponential Term

The first order of business is to get the exponential term, 3^(2x), all by its lonesome on one side of the equation. Currently, we have a pesky + 12 hanging around. To get rid of it, we'll use the golden rule of algebra: what you do to one side, you must do to the other. So, we subtract 12 from both sides of the equation:

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

This simplifies beautifully to:

3^(2x) = 27

Now, things are starting to look less cluttered, and we're one step closer to our goal.

Step 2: Expressing Both Sides with a Common Base

Here's where the magic happens. To solve for x, we need to express both sides of the equation with the same base. Why? Because if we have the same base on both sides, we can equate the exponents. It's like saying, "Hey, if these two things are equal and they have the same foundation, then their heights must be equal too!"

Currently, we have 3^(2x) on the left side. Can we express 27 as a power of 3? You bet we can! 27 is none other than 3^3. So, we rewrite the equation as:

3^(2x) = 3^3

Boom! Both sides have the same base. We're cooking with gas now!

Step 3: Equating the Exponents

Now comes the satisfying part. Since the bases are the same, we can confidently equate the exponents. This means we can drop the bases and focus solely on the powers:

2x = 3

Look at that! We've transformed a complex exponential equation into a simple linear equation. Math is beautiful, isn't it?

Step 4: Solving for x

We're in the home stretch now. To isolate x, we simply divide both sides of the equation by 2:

2x / 2 = 3 / 2

This gives us our solution:

x = 3/2

And there you have it! We've successfully navigated the equation and found the value of x. Give yourself a pat on the back – you've earned it!

Decoding the Options: Which One is the Winner?

Now that we've solved the equation, let's circle back to the multiple-choice options presented to us:

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

Comparing our solution, x = 3/2, with the options, it's clear that option C is the correct answer. We nailed it!

Why 3/2? A Deeper Dive into the Solution

We've found the answer, but let's not stop there. Let's take a moment to truly understand why x = 3/2 is the solution. We can do this by plugging our solution back into the original equation and verifying that it holds true.

Our original equation was:

3^(2x) + 12 = 39

Substituting x = 3/2, we get:

3^(2 * (3/2)) + 12 = 39

Simplifying the exponent:

3^3 + 12 = 39

We know that 3^3 = 27, so:

27 + 12 = 39

And, lo and behold:

39 = 39

The equation holds true! This confirms that x = 3/2 is indeed the correct solution. This process of verification is crucial in mathematics. It's not enough to just find an answer; we need to be sure it's the right answer.

Mastering Exponential Equations: Tips and Tricks

Solving exponential equations is a skill that gets better with practice. Here are a few tips and tricks to help you on your journey to mathematical mastery:

1. Always Isolate the Exponential Term: This is the first and most crucial step. Get that exponential term all by itself on one side of the equation.
2. Look for a Common Base: If possible, express both sides of the equation with the same base. This allows you to equate the exponents and simplify the problem.
3. Don't Forget the Logarithms: If you can't find a common base, logarithms are your best friend. They provide a powerful tool for solving exponential equations.
4. Verify Your Solution: Plugging your solution back into the original equation is a great way to catch errors and build confidence in your answer.
5. Practice, Practice, Practice: The more you practice, the more comfortable you'll become with solving exponential equations. Seek out problems of varying difficulty levels to challenge yourself.

Beyond the Equation: The Power of Mathematical Thinking

Solving equations like 3^(2x) + 12 = 39 isn't just about getting the right answer. It's about developing critical thinking skills, logical reasoning, and problem-solving abilities. These are skills that extend far beyond the realm of mathematics and can be applied to all aspects of life.

When we break down a complex problem into smaller, more manageable steps, we're not just solving an equation; we're training our minds to tackle any challenge that comes our way. So, embrace the puzzles, relish the process, and never stop exploring the fascinating world of mathematics!

In Conclusion: The Thrill of the Math Hunt

We've journeyed through the world of exponential equations, cracked the code of 3^(2x) + 12 = 39, and emerged victorious with the solution x = 3/2. But more importantly, we've learned the value of a systematic approach, the power of logical reasoning, and the satisfaction of unraveling a mathematical mystery.

So, the next time you encounter an equation that seems intimidating, remember the steps we've taken today. Remember the joy of the hunt, the thrill of the solution, and the knowledge that with a little bit of effort, you can conquer any mathematical challenge. Keep exploring, keep learning, and keep the mathematical spirit alive!

Remember guys: Mathematics is not just a subject; it's a way of thinking. And with the right tools and mindset, you can unlock its endless possibilities.