Solving Exponential Equations: Find X In 550 * 2^x = 55

Let's dive into solving the exponential equation $550 \cdot 2^x = 55$. Our goal is to isolate $x$, rounding our final answer to the nearest hundredth. Exponential equations might seem tricky at first, but with a step-by-step approach, they become quite manageable. In this article, we will explore how to tackle this specific equation and understand the underlying principles that apply to similar problems. Understanding these principles is crucial not only for academic success but also for various real-world applications, such as modeling population growth, calculating compound interest, and analyzing radioactive decay. Before we jump into the solution, it's helpful to review the basic properties of exponents and logarithms. Remember that exponents tell us how many times a base number is multiplied by itself, and logarithms are the inverse operation of exponentiation, allowing us to solve for variables within exponents. Knowing these fundamentals will make the process much smoother. Keep in mind that precision is important, especially when rounding to the nearest hundredth, as small differences can sometimes have significant impacts depending on the context of the problem. So, let's roll up our sleeves and get started on finding the value of $x$! This journey will not only enhance your problem-solving skills but also deepen your understanding of mathematical concepts, paving the way for more advanced topics in algebra and beyond. So, buckle up and get ready to unravel the mysteries of exponential equations! With each step, we'll break down the process, ensuring that you not only get the right answer but also understand why each step is necessary. Let's begin!

Step 1: Isolate the Exponential Term

The first step in solving for $x$ is to isolate the exponential term, which in this case is $2^x$. To do this, we need to get rid of the coefficient 550 that's multiplying $2^x$. We can accomplish this by dividing both sides of the equation by 550:

$550 \cdot 2^x = 55$

Divide both sides by 550:

$2^x = \frac{55}{550}$

Simplify the fraction:

$2^x = \frac{1}{10}$

Now we have the exponential term isolated, which makes it easier to proceed with solving for $x$. Isolating the variable is a common technique in algebra, and it's often the first step in solving various types of equations. This process helps simplify the equation and allows us to focus on the term that contains the variable we want to find. In this case, isolating $2^x$ sets the stage for using logarithms, which will help us bring down the exponent and solve for $x$. Remember, the goal is to manipulate the equation in a way that gets us closer to the solution while maintaining the equality. By dividing both sides by 550, we've effectively cleared the way for the next step, which involves using logarithms to unravel the exponent. So, with $2^x$ nicely isolated, we're ready to move forward and tackle the exponent. This step is crucial, as it transforms the equation into a form that's more amenable to logarithmic manipulation. Let's proceed with confidence!

Step 2: Apply Logarithms

Since we have $2^x = \frac{1}{10}$, we can now use logarithms to solve for $x$. The key idea here is to apply a logarithm to both sides of the equation. We can use any base logarithm, but the common logarithm (base 10) or the natural logarithm (base $e$) are often the most convenient because most calculators have these logarithms built-in. Let's use the common logarithm (log base 10) for this example:

$\log(2^x) = \log(\frac{1}{10})$

Now, we use the logarithmic property that allows us to bring the exponent down as a coefficient:

$x \cdot \log(2) = \log(\frac{1}{10})$

This property is a cornerstone of solving exponential equations, as it transforms the exponent into a multiplicative term, making it much easier to isolate and solve for. Remember, the logarithm of a number raised to an exponent is equal to the exponent times the logarithm of the number. This property is derived from the fundamental relationship between exponents and logarithms. By applying this property, we've effectively linearized the equation, turning an exponential problem into a linear one. This is a significant step forward, as linear equations are generally much easier to solve. Now that we have $x \cdot \log(2) = \log(\frac{1}{10})$, we can proceed to isolate $x$ by dividing both sides of the equation by $\log(2)$. This will give us the value of $x$ in terms of logarithms, which we can then evaluate using a calculator. So, with the logarithmic property in hand, we're well on our way to finding the solution! Let's keep moving forward and unravel the mysteries of this equation. The next step will bring us even closer to the answer, so stay tuned!

Step 3: Solve for x

Now that we have $x \cdot \log(2) = \log(\frac{1}{10})$, we can solve for $x$ by dividing both sides of the equation by $\log(2)$:

$x = \frac{\log(\frac{1}{10})}{\log(2)}$

We know that $\log(\frac{1}{10}) = -1$ because $10^{-1} = \frac{1}{10}$. So, we can simplify the equation:

$x = \frac{-1}{\log(2)}$

Now, we can use a calculator to find the value of $\log(2)$, which is approximately 0.30103:

$x = \frac{-1}{0.30103}$

$x \approx -3.3219$

Step 4: Round to the Nearest Hundredth

Finally, we need to round our answer to the nearest hundredth. Looking at the thousandth place (the third digit after the decimal), we see that it is 1, which is less than 5. Therefore, we round down:

$x \approx -3.32$

So, the solution to the equation $550 \cdot 2^x = 55$, rounded to the nearest hundredth, is approximately -3.32.

In conclusion, solving exponential equations involves isolating the exponential term, applying logarithms, and then using logarithmic properties to solve for the variable. Rounding to the nearest hundredth ensures that we provide an accurate and practical solution. These steps are essential for tackling a wide range of mathematical problems and real-world applications. By mastering these techniques, you'll be well-equipped to handle more complex equations and gain a deeper understanding of mathematical concepts. Remember, practice makes perfect, so keep honing your skills and exploring new challenges!

For further exploration on exponential equations and logarithms, you can visit Khan Academy's section on Exponential and Logarithmic Functions.