Use the One-to-One Property to solve the equation for x.
2 exponent x − 1 = 1/4
 

To solve the equation \( 2^x - 1 = \frac{1}{4} \) using the One-to-One Property of exponential functions, follow these steps:

1. **Isolate the exponential term**:
   \[
   2^x = \frac{1}{4} + 1
   \]
   \[
   2^x = \frac{1}{4} + \frac{4}{4} = \frac{5}{4}
   \]

2. **Apply the logarithm**: Since \( 2^x = \frac{5}{4} \), take the logarithm base 2 of both sides:
   \[
   x = \log_2\left(\frac{5}{4}\right)
   \]

3. **Use the change of base formula** if needed:
   \[
   x = \frac{\log_{10}\left(\frac{5}{4}\right)}{\log_{10}(2)}
   \]
   or
   \[
   x = \frac{\ln\left(\frac{5}{4}\right)}{\ln(2)}
   \]

Thus, the solution for \( x \) is:
\[
x = \log_2\left(\frac{5}{4}\right)
\]

### Summary
The value of \( x \) can be approximated using a calculator, but the exact form is \( \log_2\left(\frac{5}{4}\right) \).

Would you like more details on any of the steps, or do you have any questions? Here are some related questions you might find interesting:

1. What is the numerical approximation of \( \log_2\left(\frac{5}{4}\right) \)?
2. How do you convert between different logarithmic bases?
3. What are some properties of logarithms that can be useful in solving equations?
4. Can you provide examples of other equations that can be solved using the One-to-One Property?
5. How does the One-to-One Property apply to other types of functions beyond exponentials? 

**Tip**: When working with exponential equations, always try to isolate the exponential term first to make the solution process easier.

Use the One-to-One Property to solve the equation for x: 2 exponent x − 1 = 1/4.

Discover how to solve the exponential equation 2^x - 1 = 1/4 using the One-to-One Property. This guide provides step-by-step solutions and explores logarithmic applications. Ideal for high school students.

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