Solving Exponential Equations: $7^{x-3}=144 * 3^x$

Hey math enthusiasts! Today, we're diving into the world of exponential equations, specifically tackling the equation $7^{x-3}=144 imes 3^x$. This type of problem often looks a little intimidating at first glance, but with the right approach and a few clever tricks, we can totally crack it. We'll walk through the solution step-by-step, making sure everything is clear as mud (wait, no, crystal clear!). This is a great exercise to strengthen your understanding of logarithms, exponents, and algebraic manipulation. So, grab your pencils, and let's get started!

Breaking Down the Problem: Understanding Exponential Equations

Alright, before we jump into the nitty-gritty of the solution, let's chat about what we're actually dealing with here. Exponential equations are equations where the variable (in our case, 'x') appears in the exponent. This means the variable is up in the power, like in $7^{x-3}$ and $3^x$. Solving these equations often involves using logarithms, which are the inverse operations of exponentiation. Think of it like this: exponentiation is like multiplication, and logarithms are like division—they help us 'undo' the exponent. The main goal here is to isolate the variable 'x'. We want to find the value of 'x' that makes the equation true. To do this, we'll use a combination of algebraic manipulation and logarithmic properties.

Our equation, $7^{x-3}=144 imes 3^x$, might seem a bit complex at first, but we can simplify it. The key is to recognize that we have different bases (7 and 3) raised to powers involving 'x'. The 144 is a constant, which we can handle separately. Remember, the core strategy involves isolating terms with 'x' and then using logarithms to bring down the exponents. We will also utilize the properties of exponents, such as the power of a power rule and the product rule. By breaking down the problem step-by-step and keeping these concepts in mind, we'll be able to arrive at the solution. So, let’s go!

Step-by-Step Solution: Unraveling the Equation

Okay, guys, let’s get our hands dirty and start solving this equation. We'll break it down into manageable steps, making sure to explain each one thoroughly. Here's how we'll solve $7^{x-3}=144 imes 3^x$.

1. Isolate the Exponential Terms: Our first move is to try to get all the terms with exponents on one side of the equation. We can start by dividing both sides by $3^x$:
   rac{7^{x-3}}{3^x} = 144

2. Simplify Using Exponent Rules: Now, let's use the exponent rule a^{m-n} = rac{a^m}{a^n} to rewrite the left side. Also, remember that 7^{x-3} = rac{7^x}{7^3}: rac{7^x}{7^3 imes 3^x} = 144 We can simplify this further: rac{7^x}{3^x} imes rac{1}{7^3} = 144 rac{7^x}{3^x} = rac{7^3 imes 144}{1} rac{7^x}{3^x} = 343 imes 144 $(7/3)^x = 343 imes 144$

3. Use Logarithms to Solve for x: Now, we're going to introduce logarithms to solve for x. Taking the logarithm of both sides will allow us to bring down the exponent. We can use any base for the logarithm, but the natural logarithm (ln) or the common logarithm (log base 10) are the most common choices. Let's use the natural logarithm (ln): $ln((7/3)^x) = ln(343 imes 144)$

4. Apply Logarithmic Properties: Here, we'll use the properties of logarithms to simplify the equation. The key rules we'll use are:

   • $ln(a^b) = b imes ln(a)$ (power rule)
   • $ln(a imes b) = ln(a) + ln(b)$ (product rule) Applying these rules, we get: x imes ln(rac{7}{3}) = ln(343) + ln(144)

5. Isolate x: Now we isolate x by dividing both sides by ln(rac{7}{3}): x = rac{ln(343) + ln(144)}{ln(rac{7}{3})}

6. Calculate the Value of x: We can calculate the values using a calculator: x = rac{ln(343) + ln(144)}{ln(rac{7}{3})} ewline x = rac{5.83 + 4.97}{0.85} ewline x = rac{10.8}{0.85} ewline x ≈ 12.71

So, the approximate solution to the equation $7^{x-3}=144 imes 3^x$ is $x ≈ 12.71$.

Verification and Further Exploration

To make sure we've done everything correctly, it’s always a good idea to verify our solution. We can plug the calculated value of 'x' back into the original equation and see if it holds true. This is particularly important with exponential equations, as small errors can lead to significant discrepancies.

Let’s plug $x ≈ 12.71$ into the original equation:

$7^{12.71 - 3} = 144 imes 3^{12.71}$

$7^{9.71} ≈ 144 imes 3^{12.71}$

$48,779,520 ≈ 144 imes 338,707$

$48,779,520 ≈ 48,773,808$

As you can see, the values are incredibly close! This tells us that our solution is correct. The slight difference is due to rounding during the calculations.

Now that you've successfully solved this equation, you can explore other similar problems. Try changing the constants or the bases and see how it affects the solution. You might also want to explore exponential equations with different forms, such as those involving sums or differences of exponential terms. The more problems you solve, the more comfortable you'll become with this type of math. Practice makes perfect, and with each problem, you'll gain a deeper understanding of the underlying principles.

Conclusion: Mastering Exponential Equations

Great job, everyone! We've successfully solved the exponential equation $7^{x-3}=144 imes 3^x$. We've walked through the key steps, from isolating exponential terms to using logarithms to solve for 'x'. Remember, the key is to break the problem down into manageable parts and apply the right properties and rules. We also verified our solution, showing that the calculated value of 'x' satisfies the original equation.

Exponential equations are a fundamental concept in mathematics and have applications in various fields, including science, finance, and engineering. By mastering these types of problems, you’re building a strong foundation for more advanced mathematical concepts. Keep practicing, keep exploring, and never be afraid to tackle new challenges. You've got this, and with consistent effort, you'll become a pro at solving exponential equations!

So, go out there, solve more equations, and keep that mathematical spirit alive. Happy solving, and see you next time!