Hey guys! Today, we're going to dive into solving an exponential equation for the variable 't'. Exponential equations might seem intimidating at first, but with a few key steps and a solid understanding of logarithms, they become much more manageable. Let's tackle this problem together: e^(-0.55t) = 0.02
Understanding Exponential Equations
Before we jump into the solution, let's quickly recap what an exponential equation is. In its simplest form, an exponential equation involves a constant raised to a variable power. The goal is to isolate the variable, which often resides in the exponent. The equation we're dealing with, e^(-0.55t) = 0.02, features the constant 'e' (Euler's number, approximately 2.71828) as the base. To solve for 't', we need to undo the exponential operation, and that's where logarithms come into play. Logarithms are the inverse of exponential functions, meaning they help us "unwrap" the exponent and bring it down to a level where we can solve for it directly.
Why Logarithms? Logarithms provide a way to isolate the variable when it's trapped in an exponent. Think of it like this: if you have an equation like 2^x = 8, you intuitively know that x = 3 because 2 cubed is 8. But what if the number on the right side wasn't a neat power of 2? That's where logarithms shine. They give us a systematic way to find the exponent, no matter how messy the numbers get. In our case, we have e^(-0.55t) = 0.02, and we need a way to get that '-0.55t' out of the exponent.
Natural Logarithms (ln): Since our base is 'e', we'll be using the natural logarithm, denoted as 'ln'. The natural logarithm is the logarithm to the base 'e'. In other words, ln(x) answers the question: "To what power must I raise 'e' to get x?" The key property we'll use is that ln(e^x) = x. This is because the natural logarithm and the exponential function with base 'e' are inverses of each other. This property is crucial for solving our equation because it allows us to bring the exponent down.
Step-by-Step Solution
Now, let's walk through the solution step by step to make sure we understand the process completely. Remember, the goal is to isolate 't' on one side of the equation. Here's how we'll do it:
- Take the Natural Logarithm of Both Sides: To begin, we apply the natural logarithm to both sides of the equation. This is a crucial step because it allows us to use the property ln(e^x) = x to simplify the equation. So, starting with e^(-0.55t) = 0.02, we take the natural logarithm of both sides:
ln(e^(-0.55t)) = ln(0.02)
- Apply the Logarithm Property: Now we use the property ln(e^x) = x to simplify the left side of the equation. In our case, x is -0.55t. Applying the property, we get:
-0.55t = ln(0.02)
This step is where the magic happens. By taking the natural logarithm, we've successfully brought the exponent down, turning our exponential equation into a simple linear equation.
- Isolate t: To isolate 't', we need to get rid of the -0.55 that's multiplying it. We can do this by dividing both sides of the equation by -0.55:
t = ln(0.02) / -0.55
- Calculate the Value: Now, we need to calculate the value of ln(0.02) / -0.55. You'll need a calculator for this step, as ln(0.02) is not a nice, round number. Using a calculator, we find that:
ln(0.02) ≈ -3.912023
So, our equation becomes:
t ≈ -3.912023 / -0.55
Now, divide -3.912023 by -0.55:
t ≈ 7.112769
- Round to the Specified Decimal Place: It’s crucial to follow the instructions regarding rounding. If the problem specifies rounding to a certain decimal place (e.g., two decimal places), make sure to do so in the final answer. This ensures accuracy and adherence to the problem’s requirements. For instance, rounding 7.112769 to two decimal places gives us 7.11.
Final Answer
Therefore, the solution to the equation e^(-0.55t) = 0.02, rounded to two decimal places (if required), is approximately:
t ≈ 7.11
So, the correct choice would be:
A. t = 7.11
Common Mistakes to Avoid
Solving exponential equations involves several steps where errors can easily occur. Recognizing these potential pitfalls can help you avoid them and improve your accuracy. Let’s look at some common mistakes:
1. Incorrectly Applying the Logarithm Property: A frequent mistake is misapplying the logarithm property. Remember, ln(e^x) = x. It's crucial to apply the natural logarithm to both sides of the equation before attempting to simplify. Some students might try to manipulate the equation in other ways before taking the logarithm, which can lead to errors. Always ensure that you’re applying the logarithm property correctly to isolate the exponent.
2. Forgetting to Divide by the Coefficient: After applying the natural logarithm and simplifying, you'll often have an equation in the form of a constant multiplied by ‘t’ equal to some number. For example, -0.55t = ln(0.02). A common mistake is forgetting to divide both sides by the coefficient (-0.55 in this case) to isolate ‘t’. Always remember this final step of dividing to get the variable by itself.
3. Calculator Errors: Using a calculator is essential for finding the values of logarithms and performing the final calculations. However, it’s easy to make mistakes while entering the numbers. Double-check your entries, especially when dealing with negative signs and decimals. Make sure you’re using the correct function (natural logarithm ‘ln’ in this case) and that you’re following the order of operations correctly. A small mistake in calculator input can lead to a significantly wrong answer.
4. Rounding Too Early: Rounding intermediate values can introduce errors in the final answer. It’s best to keep as many decimal places as possible during the intermediate steps and only round the final answer to the specified decimal place. For example, if you round ln(0.02) before dividing by -0.55, your final answer might be slightly off. Always perform rounding as the last step to maintain accuracy.
Practice Makes Perfect
Solving exponential equations becomes easier with practice. Try working through similar problems to build your confidence and skills. Pay attention to each step, and don't hesitate to review the concepts if you get stuck. The more you practice, the more comfortable you'll become with these types of equations.
Conclusion
So, there you have it! We've successfully solved for 't' in the exponential equation e^(-0.55t) = 0.02. Remember, the key is to use logarithms to undo the exponential function and then isolate the variable. Keep practicing, and you'll become a pro at solving these types of equations. Good luck, guys, and happy solving!
