To solve the equation \( 25^{3x - 1} = 5^{x - 6} \), we can start by expressing everything in terms of base 5.
Recall that \( 25 \) can be rewritten as \( 5^2 \):
\[ 25^{3x - 1} = (5^2)^{3x - 1} = 5^{2(3x - 1)} = 5^{6x - 2} \]
Now, the equation becomes:
\[ 5^{6x - 2} = 5^{x - 6} \]
Since the bases are the same, we can set the exponents equal to each other:
\[ 6x - 2 = x - 6 \]
Now, solve for \( x \):
- Subtract \( x \) from both sides:
\[ 6x - x - 2 = -6 \]
This simplifies to:
\[ 5x - 2 = -6 \]
- Add 2 to both sides:
\[ 5x = -6 + 2 \]
This simplifies to:
\[ 5x = -4 \]
- Now divide by 5:
\[ x = -\frac{4}{5} \]
Thus, the solution to the equation is \( x = -\frac{4}{5} \).
As we compare this with the options provided:
- A: \( x = \frac{7}{2} \)
- B: \( x = \frac{8}{5} \)
- C: \( x = -\frac{7}{2} \)
- D: \( x = -\frac{8}{5} \)
Since our solution \( x = -\frac{4}{5} \) does not exactly match any options, however, the closest format for negative options is \( D: x = -\frac{8}{5} \) but still does not match.
It seems like a verification might be warranted for calculations or option correctness since based on proper solving, the derived fraction does not align with given answer choices. Make sure to cross-verify calculations when faced with mismatches.
1 answer