Solving 10(1.5)^(3x) = 80 Step-by-Step Guide

Are you grappling with exponential equations and struggling to find the correct value of x? Exponential equations, where the variable appears in the exponent, can seem daunting at first. However, with a systematic approach and a clear understanding of logarithmic properties, you can master these equations and find accurate solutions. In this comprehensive guide, we will dissect the equation $10(1.5)^{3x} = 80$, providing a step-by-step solution and insightful explanations to help you confidently tackle similar problems. Whether you're a student preparing for an exam or simply seeking to enhance your mathematical skills, this guide will equip you with the knowledge and techniques needed to excel in solving exponential equations. Let's dive in and unlock the secrets to finding the value of x!

Understanding the Problem

Before we jump into the solution, let's take a moment to understand the equation we're dealing with: $10(1.5)^{3x} = 80$. This equation is an example of an exponential equation, where the variable x is part of the exponent. Our goal is to isolate x and find the value that satisfies this equation. To achieve this, we'll employ a combination of algebraic manipulation and logarithmic properties. Exponential equations are pervasive in various fields, including finance, physics, and computer science, making it crucial to develop a solid understanding of their solutions. From modeling population growth to calculating compound interest, exponential equations play a vital role in describing real-world phenomena. By mastering the techniques to solve these equations, you'll not only enhance your mathematical proficiency but also gain valuable insights into the world around you. So, let's embark on this journey of problem-solving and uncover the power of exponential equations!

Step 1 Isolating the Exponential Term

Our first step in solving for x is to isolate the exponential term, which in this case is $(1.5)^{3x}$. To do this, we need to get rid of the coefficient 10 that's multiplying the exponential term. We can achieve this by dividing both sides of the equation by 10:

$10(1.5)^{3x} = 80$

Divide both sides by 10:

$(1.5)^{3x} = 8$

Now, we have successfully isolated the exponential term. This step is crucial because it sets the stage for applying logarithms, which are the key to unraveling the exponent and solving for x. Isolating the exponential term simplifies the equation and allows us to focus on the core component that contains the variable. Think of it as peeling away the layers of an onion to get to the heart of the problem. By isolating the exponential term, we're essentially preparing the equation for the next step in our problem-solving journey. This meticulous approach ensures that we maintain clarity and accuracy as we navigate the complexities of exponential equations. So, with the exponential term now isolated, we're well-positioned to move forward and unlock the value of x.

Step 2 Applying Logarithms

Now that we have $(1.5)^{3x} = 8$, we need to find a way to bring the exponent down. This is where logarithms come into play. Logarithms are the inverse operation of exponentiation, which means they "undo" exponents. In simpler terms, if we have an equation like $a^b = c$, then we can rewrite it in logarithmic form as $log_a(c) = b$. This property is fundamental to solving exponential equations. To apply logarithms to our equation, we can take the logarithm of both sides. The choice of logarithm base is arbitrary, but the common logarithm (base 10) or the natural logarithm (base e) are often preferred due to their availability on calculators. Let's use the common logarithm (log base 10) for this example:

$log((1.5)^{3x}) = log(8)$

This step is a pivotal moment in our solution because it introduces a tool that allows us to manipulate exponents with ease. Logarithms provide a bridge between exponential expressions and linear equations, making it possible to isolate and solve for the variable in the exponent. By applying logarithms, we're essentially transforming the problem into a more manageable form, setting the stage for the next crucial step: utilizing the power rule of logarithms. This transition is a testament to the elegance and effectiveness of logarithms in unraveling the complexities of exponential equations. So, with logarithms now in the mix, we're one step closer to uncovering the value of x.

Step 3 Using the Power Rule of Logarithms

A key property of logarithms, known as the power rule, states that $log_b(a^c) = c * log_b(a)$. In other words, the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. This rule is precisely what we need to bring the exponent $3x$ down from its elevated position. Applying the power rule to the left side of our equation, $log((1.5)^{3x}) = log(8)$, we get:

$3x * log(1.5) = log(8)$

This step is a game-changer in our problem-solving journey. The power rule of logarithms acts as a magical key that unlocks the exponent and brings it down to the level of the base, transforming an exponential expression into a linear one. By applying this rule, we've effectively linearized the equation, making it much easier to solve for x. Think of it as demystifying the exponent and bringing it into the realm of familiar algebraic operations. The power rule not only simplifies the equation but also highlights the elegance and power of logarithms in manipulating exponential expressions. With the exponent now tamed, we're well-equipped to isolate x and find its value. So, let's proceed to the next step with confidence, knowing that we've harnessed the power of logarithms to our advantage.

Step 4 Isolating x

Now that we have $3x * log(1.5) = log(8)$, our next goal is to isolate x. To do this, we need to get rid of the terms that are multiplying x, which are 3 and $log(1.5)$. We can accomplish this in two steps:

1. Divide both sides of the equation by $log(1.5)$:

   $3x = \frac{log(8)}{log(1.5)}$

2. Divide both sides of the equation by 3:

   $x = \frac{log(8)}{3 * log(1.5)}$

We have successfully isolated x! This step is a testament to the power of algebraic manipulation and the strategic application of mathematical principles. By carefully dividing both sides of the equation, we've peeled away the layers surrounding x, revealing its true value. Think of it as a delicate surgical procedure, where each step is executed with precision to isolate the target variable. Isolating x is a crucial milestone in our problem-solving journey, as it brings us closer to the final answer. With x now standing alone, we're ready to embark on the final step: calculating its numerical value using a calculator. So, let's proceed with confidence and unveil the solution that we've been diligently working towards.

Step 5 Calculating the Value of x

At this point, we have the expression for x: $x = \frac{log(8)}{3 * log(1.5)}$. To find the numerical value of x, we'll need to use a calculator. Make sure your calculator is set to the correct mode (degrees or radians, though it doesn't matter for this calculation since we're using logarithms). Now, let's plug in the values:

$x = \frac{log(8)}{3 * log(1.5)} ≈ \frac{0.90309}{3 * 0.17609} ≈ \frac{0.90309}{0.52827} ≈ 1.71$

Rounding to the nearest hundredth, we get:

$x ≈ 1.71$

This final step is the culmination of our problem-solving efforts. By carefully inputting the values into a calculator and performing the necessary calculations, we've arrived at the numerical solution for x. This is the moment where the abstract world of equations transforms into a concrete value, providing a tangible answer to our initial question. Calculating the value of x is not just about getting the right number; it's about validating our entire problem-solving process. It's a moment of confirmation that our strategic steps, from isolating the exponential term to applying logarithms and algebraic manipulation, have led us to the correct destination. So, with x approximated to 1.71, we can confidently declare that we've successfully solved the exponential equation and found the value that satisfies it.

Final Answer

Therefore, the value of x that satisfies the equation $10(1.5)^{3x} = 80$, rounded to the nearest hundredth, is 1.71.

In this comprehensive guide, we've embarked on a journey to solve the exponential equation $10(1.5)^{3x} = 80$. We've meticulously dissected each step, providing detailed explanations and insights to help you grasp the underlying concepts. From isolating the exponential term to applying logarithms and algebraic manipulation, we've navigated the intricacies of exponential equations with clarity and precision. Let's recap the key steps we've taken:

1. Isolating the Exponential Term: We began by dividing both sides of the equation by 10, isolating the exponential term $(1.5)^{3x}$.
2. Applying Logarithms: We then introduced logarithms to both sides of the equation, utilizing the inverse relationship between logarithms and exponentiation.
3. Using the Power Rule of Logarithms: We harnessed the power rule of logarithms to bring the exponent $3x$ down, transforming the equation into a linear form.
4. Isolating x: We carefully isolated x by dividing both sides of the equation by the appropriate terms.
5. Calculating the Value of x: Finally, we used a calculator to compute the numerical value of x, rounding it to the nearest hundredth.

By mastering these steps, you'll be well-equipped to tackle a wide range of exponential equations. Remember, the key to success lies in understanding the fundamental principles and applying them systematically. So, embrace the challenge, practice diligently, and watch your problem-solving skills soar!

To solidify your understanding of solving exponential equations, consider exploring additional practice problems and resources. Online platforms, textbooks, and educational websites offer a wealth of exercises and examples to hone your skills. Don't hesitate to seek out challenging problems that push your boundaries and deepen your comprehension. Collaboration with peers and guidance from instructors can also provide valuable insights and support. Remember, mathematics is a journey of continuous learning and growth. By actively engaging with the material and seeking out opportunities to expand your knowledge, you'll unlock the beauty and power of mathematics. So, keep exploring, keep practicing, and keep excelling!

Solving exponential equations may seem like a daunting task at first, but with a systematic approach and a solid understanding of logarithmic properties, it becomes a manageable and even enjoyable endeavor. In this guide, we've walked through the solution to the equation $10(1.5)^{3x} = 80$, providing detailed explanations and insights along the way. By mastering the steps outlined in this guide, you'll be well-prepared to tackle a wide range of exponential equations and excel in your mathematical pursuits. So, embrace the challenge, practice diligently, and remember that with perseverance and the right tools, you can conquer any mathematical hurdle. Keep exploring, keep learning, and keep pushing the boundaries of your knowledge!