Solving Exponential Equations: Find 'y' With Precision
Hey math enthusiasts! Ever stumbled upon an equation with an exponent and wondered how to crack the code? Today, we're diving deep into the world of exponential equations, specifically tackling the equation e7y = 12. Our mission? To isolate y and find its value, rounded to the nearest hundredth. Let's break this down step by step, making sure we don't miss a beat. Exponential equations might seem a bit intimidating at first, but once you get the hang of the rules, you'll find they're quite manageable. The key is understanding the relationship between exponents and logarithms, which are essentially inverse operations. When you have an exponential equation, the goal is always to get the variable out of the exponent and onto its own. This is where logarithms come into play, providing the perfect tool for the job. We'll use the natural logarithm (ln), which is the logarithm with base e (Euler's number, approximately 2.71828). It is the most common log, so most calculators have a button for it. So, let's get started on solving for y.
First, we want to isolate the exponent. In our equation, e7y = 12, the exponential part is already isolated on one side of the equation. This is the ideal starting point! The next step is to use the natural logarithm on both sides. Applying the natural logarithm to both sides of the equation maintains the equality, which is a fundamental rule in mathematics. This means we'll take the natural logarithm (ln) of both sides of the equation. When we do this, the equation becomes ln(e7y) = ln(12). The beauty of this step is that the natural logarithm and the exponential function with base e are inverses of each other. This means that ln(ex) = x. This property will help us to simplify our equation significantly. This is a very common technique in solving exponential equations, because the properties of logarithms allow you to 'bring down' the exponent. This will allow you to get the variable y out of the exponent and solve for it like a regular algebra problem.
Next, apply the power rule of logarithms. The power rule of logarithms states that ln(ab) = b * ln(a). In our equation, we can use this rule to bring the exponent down. In our case, 7y comes down. Applying the power rule to ln(e7y) = ln(12), we get 7y * ln(e) = ln(12). Remember that ln(e) is equal to 1, because the natural log is base e, and the log of its base is always 1. So, this simplifies our equation to 7y * 1 = ln(12), or just 7y = ln(12). Now, we're getting closer to solving for y. This step is crucial because it allows us to 'peel off' the exponent and work directly with y. The power rule is a lifesaver in these scenarios, and it's something you'll use regularly when working with exponential equations. The more practice you get with this, the more easily you will recognize when to use it, and you'll find that it greatly simplifies the problem.
Now, solve for y. We have the simplified equation 7y = ln(12). To isolate y, we need to divide both sides of the equation by 7. Doing this gives us y = ln(12) / 7. At this point, you'll need a calculator. Make sure your calculator has the natural log function (usually labeled as
