If `25^(x-1)=5^(2x-1)-100`, then find the value of `x`.

Explanation

To solve the equation \( 25^{(x-1)} = 5^{(2x-1)} - 100 \), we can follow these steps:

### Step 1: Rewrite the bases
First, we can express \( 25 \) in terms of \( 5 \):
\[ 25 = 5^2 \]
Thus, we can rewrite \( 25^{(x-1)} \) as:
\[ (5^2)^{(x-1)} = 5^{2(x-1)} = 5^{(2x-2)} \]
Now, our equation becomes:
\[ 5^{(2x-2)} = 5^{(2x-1)} - 100 \]

### Step 2: Rearrange the equation
Next, we can rearrange the equation to bring all terms involving \( 5 \) to one side:
\[ 5^{(2x-2)} - 5^{(2x-1)} = -100 \]

### Step 3: Factor out common terms
Notice that \( 5^{(2x-2)} \) can be factored out:
\[ 5^{(2x-2)} (1 - 5) = -100 \]
This simplifies to:
\[ 5^{(2x-2)} (-4) = -100 \]
Dividing both sides by \(-4\) gives:
\[ 5^{(2x-2)} = \frac{100}{4} = 25 \]

### Step 4: Rewrite \( 25 \) as a power of \( 5 \)
Now, we can express \( 25 \) as \( 5^2 \):
\[ 5^{(2x-2)} = 5^2 \]

### Step 5: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:
\[ 2x - 2 = 2 \]

### Step 6: Solve for \( x \)
Now, we can solve for \( x \):
\[ 2x = 2 + 2 \]
\[ 2x = 4 \]
\[ x = \frac{4}{2} = 2 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{2} \]

---