To solve the equation \( 50 = e^{-5600k + 4.605} \) for \( k \), follow these steps:

### Step 1: Take the natural logarithm (ln) on both sides
\[
\ln(50) = \ln\left(e^{-5600k + 4.605}\right)
\]
Since the natural logarithm and the exponential function are inverse functions:
\[
\ln(50) = -5600k + 4.605
\]

### Step 2: Isolate the term with \( k \)
Subtract 4.605 from both sides:
\[
\ln(50) - 4.605 = -5600k
\]

### Step 3: Solve for \( k \)
Divide both sides by \(-5600\):
\[
k = \frac{\ln(50) - 4.605}{-5600}
\]

### Step 4: Calculate the value
Now, compute the value using a calculator:
\[
\ln(50) \approx 3.912
\]
\[
k = \frac{3.912 - 4.605}{-5600} = \frac{-0.693}{-5600} \approx 1.2375 \times 10^{-4}
\]

Thus, the value of \( k \) is approximately \( 1.2375 \times 10^{-4} \).

Would you like more details or have any questions? Here are five related questions:

1. What is the significance of the natural logarithm in solving exponential equations?
2. How can we verify the solution of \( k \) by substituting it back into the original equation?
3. What are some common uses of exponential decay formulas in real-world scenarios?
4. How would the equation change if the constant added to the exponent was different?
5. What is the effect of changing the coefficient of \( k \) in the exponent on the value of \( k \)?

**Tip:** When working with exponential equations, always remember that taking the natural logarithm of both sides is a powerful tool to isolate the exponent.

Learn how to solve the exponential equation 50 = e^(-5600k + 4.605) step-by-step. This problem involves using the natural logarithm to isolate the exponent and calculate the value of k. Suitable for advanced high school students and anyone interested in exponential equations.

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