The equation `(0.4)^(x-1)=(6.25)^(6x-5)` has

Options

A. no solution
B. one solution
C. two solutions
D. more than two solutions

Explanation

To solve the equation \( (0.4)^{x-1} = (6.25)^{6x-5} \), we will follow these steps:

### Step 1: Rewrite the bases
First, we can express \( 0.4 \) and \( 6.25 \) in terms of fractions:
\[ 0.4 = \frac{4}{10} = \frac{2}{5} \]
\[ 6.25 = \frac{625}{100} = \frac{25}{4} = \left(\frac{5}{2}\right)^2 \]
Thus, we can rewrite the equation as:
\[ \left(\frac{2}{5}\right)^{x-1} = \left(\frac{5}{2}\right)^{2(6x-5)} \]

### Step 2: Simplify the right side
The right side can be simplified:
\[ \left(\frac{5}{2}\right)^{2(6x-5)} = \left(\frac{5}{2}\right)^{12x - 10} \]
So, we have:
\[ \left(\frac{2}{5}\right)^{x-1} = \left(\frac{5}{2}\right)^{12x - 10} \]

### Step 3: Rewrite the left side
We can rewrite the left side as:
\[ \left(\frac{2}{5}\right)^{x-1} = \left(\frac{5}{2}\right)^{-(x-1)} \]
Thus, we have:
\[ \left(\frac{5}{2}\right)^{-(x-1)} = \left(\frac{5}{2}\right)^{12x - 10} \]

### Step 4: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:
\[ -(x-1) = 12x - 10 \]

### Step 5: Solve for \( x \)
Now, we simplify the equation:
\[ -x + 1 = 12x - 10 \]
Adding \( x \) to both sides:
\[ 1 = 13x - 10 \]
Adding \( 10 \) to both sides:
\[ 11 = 13x \]
Dividing both sides by \( 13 \):
\[ x = \frac{11}{13} \]

### Conclusion
The equation \( (0.4)^{x-1} = (6.25)^{6x-5} \) has one solution, which is:
\[ x = \frac{11}{13} \]