https://file.aiquickdraw.com/mathbot/file/1725495422468tx52yysn.jpg

The equation shown in the image is:

\[
e^{x - 1} = 2
\]

To solve for \( x \), follow these steps:

1. Take the natural logarithm (ln) on both sides to eliminate the exponential function:

\[
\ln(e^{x - 1}) = \ln(2)
\]

2. Simplify the left side using the property \( \ln(e^a) = a \):

\[
x - 1 = \ln(2)
\]

3. Solve for \( x \) by adding 1 to both sides:

\[
x = \ln(2) + 1
\]

You can leave the answer in this form or evaluate it using an approximate value for \( \ln(2) \approx 0.693 \):

\[
x \approx 0.693 + 1 = 1.693
\]

Thus, \( x \approx 1.693 \).

Would you like more details, or have any other questions?

Here are some related questions:
1. How do logarithmic properties help in solving exponential equations?
2. What is the significance of the natural logarithm (\( \ln \)) in mathematics?
3. How does this method apply to different bases, such as \( 2^{x-1} = 2 \)?
4. Can this equation have other types of solutions if additional terms were included?
5. How would solving this equation change if the right-hand side were negative?

**Tip:** When dealing with equations involving exponentials and logarithms, applying the natural logarithm is often a helpful first step to simplify the equation.

Learn how to solve the exponential equation e^(x-1) = 2 by applying logarithmic properties. This step-by-step solution covers essential mathematical concepts like exponential equations and logarithms, suitable for students in Grades 10-12.

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