To linearize the equation \( x = e^{\frac{y-b}{a}} \), we can take the natural logarithm of both sides. Here's the step-by-step process:

Given:
\[
x = e^{\frac{y-b}{a}}
\]

1. **Take the natural logarithm of both sides**:
   \[
   \ln(x) = \ln\left(e^{\frac{y-b}{a}}\right)
   \]

2. **Simplify the right-hand side** using the logarithm property \( \ln(e^u) = u \):
   \[
   \ln(x) = \frac{y-b}{a}
   \]

3. **Multiply both sides by \( a \)** to solve for \( y \):
   \[
   a \ln(x) = y - b
   \]

4. **Add \( b \) to both sides** to isolate \( y \):
   \[
   y = a \ln(x) + b
   \]

The linearized form of the equation is:
\[
y = a \ln(x) + b
\]

This equation is now in a linear form, where \( \ln(x) \) is the independent variable, and \( y \) is the dependent variable.

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How do you find the slope \( a \) from experimental data if you have the values of \( x \) and \( y \)?
2. What is the significance of the intercept \( b \) in the linearized equation?
3. How would you plot this linearized equation on a graph?
4. Can you linearize other exponential functions in a similar way?
5. How does linearization help in analyzing nonlinear relationships?

**Tip:** When linearizing a function, always identify the part of the equation that can be transformed into a linear term, such as using logarithms for exponential functions.

Learn how to linearize the exponential equation x = e^((y-b)/a) using logarithms step-by-step. This problem involves exponential functions, logarithms, and algebraic manipulation. Suitable for advanced high school students and above.

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