Solving Exponential Equations: Find X In 7^(-x+8) = 8^(-4x)

Hey there, math enthusiasts! Today, we're diving into the exciting world of exponential equations to solve for x in the equation 7^(-x+8) = 8^(-4x). Don't worry if it looks intimidating at first. We'll break it down step by step, making sure to round our final answer to the nearest thousandth. Let's get started!

Understanding Exponential Equations

Before we jump into the solution, let's quickly recap what exponential equations are all about. An exponential equation is one where the variable appears in the exponent. These types of equations often require us to use logarithms to isolate the variable. The key idea here is that logarithms are the inverse operation of exponentiation. This means that if we have an equation where a variable is in the exponent, applying a logarithm to both sides can help us bring that exponent down and solve for the variable. The beauty of logarithms lies in their ability to transform complex exponential problems into simpler algebraic ones. We will leverage the properties of logarithms to solve our equation, making the process much more manageable.

Why Logarithms Are Our Best Friend

Think of logarithms as the superpower we need to tackle exponential equations. When we have an equation like 7^(-x+8) = 8^(-4x), where x is hanging out in the exponent, we need a way to bring it down to ground level. That's where logarithms come in. By taking the logarithm of both sides of the equation, we can use the power rule of logarithms, which states that log_b(a^c) = c * log_b(a). This rule allows us to move the exponent (-x + 8) and (-4x) from their elevated positions down to become coefficients, making the equation much easier to handle. In essence, logarithms provide us with a tool to unravel the exponential relationship and transform it into a linear one, which we can then solve using basic algebraic techniques. This is why understanding logarithms is crucial for anyone delving into exponential equations.

Properties of Logarithms

To effectively use logarithms, it’s essential to understand some of their key properties. One of the most important is the power rule, which we’ve already touched upon. This rule is a game-changer when dealing with exponential equations because it allows us to move exponents out in front as coefficients. Another crucial property is the product rule, which states that log_b(mn) = log_b(m) + log_b(n). This rule is handy when we have products inside a logarithm. Similarly, the quotient rule, log_b(m/n) = log_b(m) - log_b(n), helps us deal with quotients. These properties are like the secret ingredients in our mathematical toolkit, allowing us to manipulate and simplify complex expressions. By mastering these rules, we can confidently tackle a wide range of logarithmic and exponential equations, turning what might seem like daunting problems into straightforward exercises.

Step-by-Step Solution

Now, let's get our hands dirty and solve the equation 7^(-x+8) = 8^(-4x) step by step.

Step 1: Apply the Natural Logarithm to Both Sides

To start, we'll apply the natural logarithm (ln) to both sides of the equation. Why the natural logarithm? Well, it's a common choice and works perfectly for this situation. The natural logarithm has a base of e, which is a special number in mathematics (approximately 2.71828). Applying the natural logarithm to both sides gives us:

ln(7^(-x+8)) = ln(8^(-4x))

Step 2: Use the Power Rule of Logarithms

The power rule of logarithms states that ln(a^b) = b * ln(a). We can use this rule to bring the exponents down:

(-x + 8) * ln(7) = -4x * ln(8)

This step is crucial because it transforms the exponential equation into a linear one, which is much easier to solve. By applying the power rule, we’ve effectively removed the variable from the exponent, making it a coefficient instead. This allows us to manipulate the equation using basic algebraic techniques. Now, we have a straightforward linear equation involving x, which we can solve by isolating x on one side. This step showcases the power of logarithms in simplifying complex exponential equations.

Step 3: Distribute and Expand

Next, we need to distribute ln(7) on the left side of the equation:

-x * ln(7) + 8 * ln(7) = -4x * ln(8)

Step 4: Rearrange the Equation

Let's move all the terms containing x to one side and the constants to the other side. Add x * ln(7) to both sides:

8 * ln(7) = -4x * ln(8) + x * ln(7)

Step 5: Factor out x

Now, factor out x from the right side:

8 * ln(7) = x * (-4 * ln(8) + ln(7))

Step 6: Isolate x

To isolate x, divide both sides by (-4 * ln(8) + ln(7)):

x = (8 * ln(7)) / (-4 * ln(8) + ln(7))

This step is the culmination of our efforts to isolate x. By dividing both sides of the equation by the expression in parentheses, we finally have x standing alone on one side. The expression on the other side gives us the exact value of x, which we can then approximate using a calculator. This isolation of x is the ultimate goal when solving any equation, and it’s a testament to the power of algebraic manipulation. The journey from the original exponential equation to this point has been a testament to the power of logarithms and algebraic techniques in unraveling mathematical problems.

Step 7: Calculate the Value of x

Using a calculator, we can find the approximate value of x:

x ≈ (8 * 1.9459) / (-4 * 2.0794 + 1.9459) x ≈ 15.5672 / (-8.3176 + 1.9459) x ≈ 15.5672 / -6.3717 x ≈ -2.443

Step 8: Round to the Nearest Thousandth

Rounding our answer to the nearest thousandth, we get:

x ≈ -2.443

Conclusion

So, the solution to the equation 7^(-x+8) = 8^(-4x), rounded to the nearest thousandth, is approximately -2.443. Solving exponential equations might seem tricky at first, but with a good understanding of logarithms and their properties, you can tackle them with confidence. Remember to apply the natural logarithm, use the power rule, and carefully isolate the variable. Keep practicing, and you'll become a pro at solving these types of equations!

For further exploration of exponential equations and logarithms, you might find valuable resources and explanations on websites like Khan Academy, which offers a wealth of educational content on mathematics.