how do i solve the equation x= 1/50(1-e^-2/5) please explain the comp method

To solve the equation \( x = \frac{1}{50}(1 - e^{-2/5}) \), we can break it down step by step. This equation involves an exponential function, which can often seem tricky at first, but with a clear approach, it becomes manageable. Let's go through the process together.

Understanding the Components

First, let's identify the components of the equation:

• x: This is the variable we are trying to solve for.
• e: This is the base of the natural logarithm, approximately equal to 2.71828.
• -2/5: This exponent indicates how the exponential function behaves.

Step-by-Step Solution

Now, let's solve the equation step by step:

1. Isolate the Exponential Term

We start with the equation:

x = \frac{1}{50}(1 - e^{-2/5})

To isolate the exponential term, we can rearrange the equation. First, multiply both sides by 50:

50x = 1 - e^{-2/5}

2. Rearranging for e

Next, we want to isolate \( e^{-2/5} \). We can do this by rearranging the equation:

e^{-2/5} = 1 - 50x

3. Exponentiation

To eliminate the negative exponent, we can take the reciprocal:

e^{2/5} = \frac{1}{1 - 50x}

4. Taking the Natural Logarithm

Now, we can take the natural logarithm of both sides to solve for \( -\frac{2}{5} \):

-2/5 = \ln\left(\frac{1}{1 - 50x}\right)

5. Solving for x

Finally, we can solve for \( x \) by rearranging the equation again. First, multiply both sides by -5/2:

x = \frac{1}{50} \left(1 - e^{-2/5}\right)

Example Calculation

Let’s say we want to find \( x \) when we substitute a specific value. For instance, if \( x = 0.01 \):

We can plug this value back into our equation:

0.01 = \frac{1}{50}(1 - e^{-2/5})

From here, you can calculate \( e^{-2/5} \) using a calculator or software, and then check if both sides of the equation balance.

Final Thoughts

By following these steps, you can systematically solve equations involving exponential functions. Remember, practice is key to becoming comfortable with these types of problems. If you have any further questions or need clarification on any step, feel free to ask!