We are given the equation:

\[
14e^{5x} + 2 = 560
\]

We can solve for \( x \) by following these steps:

### Step 1: Isolate the exponential term
First, subtract 2 from both sides to get the exponential term by itself:

\[
14e^{5x} = 560 - 2
\]

\[
14e^{5x} = 558
\]

### Step 2: Divide by 14
Now, divide both sides by 14 to isolate \( e^{5x} \):

\[
e^{5x} = \frac{558}{14}
\]

\[
e^{5x} = 39.857
\]

### Step 3: Take the natural logarithm (ln) of both sides
To eliminate the exponential, take the natural logarithm of both sides:

\[
\ln(e^{5x}) = \ln(39.857)
\]

Using the logarithmic identity \( \ln(e^y) = y \), we get:

\[
5x = \ln(39.857)
\]

\[
5x = 3.684
\]

### Step 4: Solve for \( x \)
Finally, divide both sides by 5:

\[
x = \frac{3.684}{5}
\]

\[
x = 0.737
\]

Thus, the solution is:

\[
x \approx 0.737
\]

Would you like more details on any of these steps?

### 5 Related Questions:
1. How do you solve equations with logarithms and exponents?
2. What is the natural logarithm and how is it used in solving exponential equations?
3. Can exponential functions model real-life scenarios?
4. What are common properties of the natural base \( e \)?
5. How do we interpret solutions of equations involving \( e \) in applied contexts?

### Tip:
When dealing with exponential equations, remember that taking the natural logarithm is often the best way to simplify terms involving \( e \).

Learn how to solve the exponential equation 14e^5x + 2 = 560 with a step-by-step solution. This problem covers key concepts of exponential equations and logarithms, including the use of natural logarithms to simplify and solve equations. Suitable for high school students in Grades 10-12.

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