The value of x if `5^(2x-1)=25^(x-1)+100` is __________.

Options

A. 8
B. 5
C. 2
D. 0

Answer

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

### Step 1: Rewrite the equation
We know that \( 25 \) can be expressed as \( 5^2 \). Therefore, we can rewrite \( 25^{x-1} \) as:
\[ 25^{x-1} = (5^2)^{x-1} = 5^{2(x-1)} = 5^{2x - 2} \]
Now, substituting this back into the equation gives us:
\[ 5^{2x-1} = 5^{2x-2} + 100 \]

### Step 2: Simplify the equation
Next, we can rewrite the equation as:
\[ 5^{2x-1} - 5^{2x-2} = 100 \]
Factoring out \( 5^{2x-2} \) from the left side:
\[ 5^{2x-2}(5 - 1) = 100 \]
This simplifies to:
\[ 5^{2x-2} \cdot 4 = 100 \]

### Step 3: Divide both sides by 4
Now, we can divide both sides by 4:
\[ 5^{2x-2} = \frac{100}{4} = 25 \]

### Step 4: Rewrite 25 in terms of base 5
Since \( 25 = 5^2 \), we can rewrite the equation as:
\[ 5^{2x-2} = 5^2 \]

### Step 5: Set the exponents equal to each other
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 \( x \):
\[ 2x = 2 + 2 \]
\[ 2x = 4 \]
\[ x = \frac{4}{2} = 2 \]

Thus, the value of \( x \) is \( \boxed{2} \).

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