We are given the logarithmic equation:

\[
\log_{0.25} (2x - 1) = -1
\]

### Step 1: Convert the logarithmic equation to its exponential form

The general form of a logarithmic equation is:

\[
\log_b (y) = x \implies y = b^x
\]

In our case, \( b = 0.25 \), \( y = 2x - 1 \), and \( x = -1 \). Applying the exponential rule, we get:

\[
2x - 1 = 0.25^{-1}
\]

### Step 2: Simplify the exponential expression

\[
0.25^{-1} = \frac{1}{0.25} = 4
\]

So, we now have:

\[
2x - 1 = 4
\]

### Step 3: Solve for \( x \)

Add 1 to both sides:

\[
2x = 4 + 1 = 5
\]

Now divide by 2:

\[
x = \frac{5}{2} = 2.5
\]

### Final Solution:

\[
x = 2.5
\]

Would you like more details or have any questions?

Here are 5 related questions you might find useful:
1. How do you convert logarithms with fractional bases into exponential form?
2. What are the properties of logarithms with bases less than 1?
3. How do you solve exponential equations involving fractional bases?
4. Can logarithmic equations have negative solutions, and if so, when?
5. What are the common mistakes when dealing with logarithms and exponents?

**Tip:** When working with logarithms, always make sure the argument inside the logarithm is positive!

Learn how to solve the logarithmic equation log 0.25 (2x − 1) = −1 by converting it into an exponential equation. This problem covers key concepts like logarithmic to exponential conversion and solving for variables in algebraic equations. Suitable for students in Grades 9-11.

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