Question

Solve the following exponential equation: 25^3x-1=5^x-6



A
x=72x=\frac{7}{2}x=
2
7
​


B
x=85x=\frac{8}{5}x=
5
8
​


C
x=−72x=-\frac{7}{2}x=−
2
7
​


D
x=−85x=-\frac{8}{5}x=−
5
8
​

Answer 1

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 \( x \):

1. Subtract \( x \) from both sides:

\[
6x - x - 2 = -6
\]

This simplifies to:

\[
5x - 2 = -6
\]

2. Add 2 to both sides:

\[
5x = -6 + 2
\]

This simplifies to:

\[
5x = -4
\]

3. 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.