Solving Exponential Equations: Step-by-Step Guide

Hey everyone! Today, we're diving into the world of exponential equations. Specifically, we're going to solve for x in the equation $7^{-x-4} = 9^{-6x}$. It might seem a little intimidating at first, but trust me, we'll break it down step by step and make it super clear. We'll use logarithms to isolate x and get our final answer. So, grab your calculators, and let's get started. We'll also make sure to round our answer to the nearest thousandth as requested.

Understanding Exponential Equations and Logarithms

Before we jump into the problem, let's quickly recap what exponential equations and logarithms are all about. Exponential equations are equations where the variable appears in the exponent. For instance, $2^x = 8$ is an exponential equation. Solving these equations often involves using logarithms. A logarithm is the inverse operation of exponentiation. Basically, it answers the question: "To what power must we raise the base to get a certain number?" For example, $\log_2 8 = 3$ because $2^3 = 8$. The logarithm tells us the exponent. This relationship is key to solving exponential equations. The main thing is to use logarithms to bring down the exponents so that we can solve for $x$ easily. Now, we are prepared to solve the equation $7^{-x-4} = 9^{-6x}$.

Let's get into the details. The core idea is to manipulate the equation using logarithmic properties until we can isolate x. We'll apply the logarithm to both sides of the equation, which doesn't change the equality. There are different types of logarithms that can be used, such as common logarithms (base 10) or natural logarithms (base e). For this problem, we can use any of them, but I will be using the natural logarithm (ln) because it is widely used. The key is to keep the equation balanced, applying the same operation to both sides. As we progress, we'll simplify and rearrange the terms, collecting all the x terms on one side and the constants on the other. This process is all about making the equation easier to solve until we can find the value of x. Remember that practice makes perfect, so don't be discouraged if it takes a few tries to fully grasp the concepts. Let's do it!

Alright, let's get our hands dirty with the equation. Our equation is $7^{-x-4} = 9^{-6x}$. The first step is to take the natural logarithm (ln) of both sides. This gives us $\ln(7^{-x-4}) = \ln(9^{-6x})$. Using the power rule of logarithms, we can bring the exponents down: $(-x-4)\ln(7) = (-6x)\ln(9)$. Now, let's distribute the $\ln(7)$ on the left side: $-x\ln(7) - 4\ln(7) = -6x\ln(9)$. Our goal is to isolate x. Let's move all the terms containing x to one side and the constant terms to the other side. Add $6x\ln(9)$ to both sides: $-x\ln(7) + 6x\ln(9) - 4\ln(7) = 0$. Then, add $4\ln(7)$ to both sides: $-x\ln(7) + 6x\ln(9) = 4\ln(7)$. Now, we factor out x on the left side: $x(- \ln(7) + 6\ln(9)) = 4\ln(7)$. Finally, to solve for x, divide both sides by $(- \ln(7) + 6\ln(9))$: $x = \frac{4\ln(7)}{-\ln(7) + 6\ln(9)}$. This is our solution for x. We are on the right path, guys!

Detailed Step-by-Step Solution

Now, let's get into the step-by-step solution to make everything clear. The original equation is $7^{-x-4} = 9^{-6x}$. We want to find the value of x that satisfies this equation. Here's how we're going to do it:

1. Take the natural logarithm (ln) of both sides: $\ln(7^{-x-4}) = \ln(9^{-6x})$. This step allows us to use the power rule of logarithms.
2. Apply the power rule: Using the property $\ln(a^b) = b\ln(a)$, we get: $(-x-4)\ln(7) = -6x\ln(9)$. This rule is super important because it brings the exponents down, making the equation easier to manipulate.
3. Expand the left side: Distribute $\ln(7)$: $-x\ln(7) - 4\ln(7) = -6x\ln(9)$. This step simplifies the left side of the equation.
4. Rearrange the equation to group x terms: Add $6x\ln(9)$ to both sides and add $4\ln(7)$ to both sides. This gives us $6x\ln(9) - x\ln(7) = 4\ln(7)$. The objective is to bring all the x terms to one side.
5. Factor out x: Factor x from the left side: $x(6\ln(9) - \ln(7)) = 4\ln(7)$. Factoring x allows us to isolate it more easily.
6. Solve for x: Divide both sides by $(6\ln(9) - \ln(7))$: $x = \frac{4\ln(7)}{6\ln(9) - \ln(7)}$. We've now isolated x and have an expression that we can evaluate.

Now, let's use a calculator to find the numerical value of x. Guys, make sure you know how to use your calculator. Using a calculator, we find that $\ln(7) \approx 1.94591$ and $\ln(9) \approx 2.19722$. Plugging these values into the equation, we get: $x = \frac{4(1.94591)}{6(2.19722) - 1.94591} = \frac{7.78364}{13.18332 - 1.94591} = \frac{7.78364}{11.23741} \approx 0.6926$. Therefore, the solution for x is approximately 0.693, rounded to the nearest thousandth. Remember to always double-check your work and make sure you understand each step. This process is applicable to similar exponential equations, so understanding it well will help you solve more complex problems in the future. Keep practicing, and you'll become a pro at solving these types of equations in no time! Woohoo!

Calculator Time and Final Answer

Alright, let's calculate the final answer. We've got the equation $x = \frac{4\ln(7)}{6\ln(9) - \ln(7)}$. We'll use a calculator to find the values of $\ln(7)$ and $\ln(9)$.

• $\ln(7) \approx 1.94591014906$
• $\ln(9) \approx 2.19722457734$

Now, substitute these values into the equation: $x = \frac{4 \times 1.94591014906}{6 \times 2.19722457734 - 1.94591014906} = \frac{7.78364059624}{13.183347464 - 1.94591014906} = \frac{7.78364059624}{11.2374373149} \approx 0.69265$. When we round our answer to the nearest thousandth, we get $x \approx 0.693$.

So, there you have it, folks! We've successfully solved for x in the equation $7^{-x-4} = 9^{-6x}$, and the answer, rounded to the nearest thousandth, is approximately 0.693. Always remember to double-check your work, and don't be afraid to practice. The more you practice, the easier it gets. Exponential equations might seem tricky at first, but once you break them down into steps, they become much more manageable. Keep exploring and keep learning. This is how you master mathematics! Guys, you are amazing!