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

Question

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 \( 25^{3x+1} = 5^{x-6} \), let's first express everything in terms of the same base.

Since \( 25 \) can be written as \( 5^2 \), we have:

\[ 25^{3x+1} = (5^2)^{3x+1} = 5^{2(3x+1)} = 5^{6x + 2} \]

Now substituting back into the original equation gives us:

\[ 5^{6x + 2} = 5^{x - 6} \]

Since the bases are the same, we can set the exponents equal to each other:

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

Now, we will solve for \( x \):

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

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

This simplifies to:

\[ 5x + 2 = -6 \]

2. Next, subtract \( 2 \) from both sides:

\[ 5x = -6 - 2 \]

This results in:

\[ 5x = -8 \]

3. Now, divide by \( 5 \):

\[ x = -\frac{8}{5} \]

Thus, the solution is:

\[ x = -\frac{8}{5} \]

The answer corresponds to option D:

D \(x = -\frac{8}{5}\).