Solving For X In 17^(3x) = 18^(x-3) A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into an exciting exponential equation to solve for x. Our equation is 17^(3x) = 18^(x-3). This might look a bit intimidating at first, but don't worry, we'll break it down step by step and use logarithms to find the exact solution. So, let’s put on our thinking caps and get started!

Understanding Exponential Equations

Before we jump into solving, let's quickly recap what exponential equations are. Exponential equations are equations where the variable appears in the exponent. They often involve a constant base raised to a power that includes the variable, like our 17^(3x) and 18^(x-3). The key to solving these equations is to manipulate them in a way that we can isolate the variable. This often involves using logarithms, which are the inverse operations of exponentiation. Logarithms allow us to bring the exponents down and solve for the variable more easily. Remember, the basic idea behind using logarithms is that if we have an equation like a^b = c, we can rewrite it in logarithmic form as log_a(c) = b. This transformation is crucial for solving equations where the variable is in the exponent.

In our case, we have two exponential terms with different bases, 17 and 18. This adds a layer of complexity, but it also means we'll get to use some cool logarithmic properties to simplify things. We'll be using properties like log(a^b) = b*log(a) to bring those exponents down and rearrange the equation. So, keep this in mind as we move forward. The goal is always to get x by itself, and logarithms are our best friend in this endeavor. We’ll be working with both sides of the equation, applying the same operations to maintain balance, which is a fundamental principle in algebra. So, let's get our logarithmic tools ready and tackle this problem!

Applying Logarithms to Both Sides

Okay, guys, the first crucial step in solving this exponential equation is to apply a logarithm to both sides. This might sound like a big leap, but trust me, it's the key to unlocking the solution. When we apply a logarithm, we're essentially using the property that if a = b, then log(a) = log(b). This maintains the equality of the equation while allowing us to manipulate the exponents more easily. You might be wondering, which logarithm should we use? Well, we have a couple of options: base-10 logarithms (log) or natural logarithms (ln, which is base-e). Both will work perfectly fine, and the choice often comes down to personal preference or what your calculator handles most easily. For this example, let's go ahead and use the natural logarithm (ln) because it’s quite common and handy.

So, we start with our equation: 17^(3x) = 18^(x-3). Now, we apply the natural logarithm to both sides: ln(17^(3x)) = ln(18^(x-3)). See what we did there? We're keeping both sides balanced by performing the same operation. This is a fundamental rule in algebra, ensuring that our equation remains valid as we manipulate it. Now, here comes the fun part – using the power rule of logarithms! This rule states that ln(a^b) = b * ln(a). Applying this rule to both sides of our equation is what allows us to bring the exponents down. This is a game-changer because it turns our exponents, which contain the variable x, into coefficients, making them much easier to work with. So, get ready to see how this logarithmic magic unfolds as we move on to the next step.

Using the Power Rule of Logarithms

Alright, now let's wield the power rule of logarithms like pros! As we mentioned earlier, the power rule states that ln(a^b) = b * ln(a). This is a crucial property that helps us bring down the exponents in our equation, making it much easier to solve for x. We've already applied the natural logarithm to both sides of our equation, giving us ln(17^(3x)) = ln(18^(x-3)). Now, it’s time to put the power rule into action.

On the left side, we have ln(17^(3x)). Applying the power rule, we can rewrite this as 3x * ln(17). Notice how the exponent 3x has now become a coefficient, multiplying the natural logarithm of 17. Similarly, on the right side, we have ln(18^(x-3)). Applying the power rule here, we get (x - 3) * ln(18). Again, the exponent (x - 3) has come down and is now multiplying the natural logarithm of 18. So, our equation now looks like this: 3x * ln(17) = (x - 3) * ln(18). See how much simpler it looks already? By using the power rule, we've transformed the exponential equation into a linear equation involving x, which is something we can handle much more easily. Now, we just need to keep going, applying more algebraic techniques to isolate x. Let’s keep pushing forward!

Distributing and Rearranging the Equation

Okay, team, we've successfully used the power rule to bring down those exponents, and now our equation looks like 3x * ln(17) = (x - 3) * ln(18). The next step is to distribute and rearrange the equation so we can group the x terms together. This is a standard algebraic technique that’s super important for solving equations. We need to get all the terms involving x on one side of the equation and the constant terms on the other side.

First, let’s distribute the ln(18) on the right side. This means we’ll multiply ln(18) by both x and -3. So, (x - 3) * ln(18) becomes x * ln(18) - 3 * ln(18). Now our equation looks like this: 3x * ln(17) = x * ln(18) - 3 * ln(18). Great! We’ve expanded the equation and gotten rid of the parentheses. Now, we want to get all the x terms on one side. Let's subtract x * ln(18) from both sides. This gives us: 3x * ln(17) - x * ln(18) = -3 * ln(18). We’re getting closer! Now we have all the x terms on the left side and the constant term on the right side. The next step is to factor out x from the left side, which will help us isolate x completely. So, stay with me, we're almost there!

Factoring Out x

Alright, let's keep the momentum going! We've reached the point where our equation looks like 3x * ln(17) - x * ln(18) = -3 * ln(18). Now, the key step here is to factor out x from the left side of the equation. Factoring is a fundamental algebraic technique that allows us to simplify expressions and isolate variables. In this case, both terms on the left side have x in them, so we can factor it out.

When we factor out x, we're essentially reversing the distributive property. We're saying,