Solve for x 5^(8(x-1))=625^(2x-2)

$5^{8 (x - 1)} = 625^{2 x - 2}$

Step 1

Create equivalent expressions in the equation that all have equal bases.

$5^{8 (x - 1)} = 5^{4 (2 x - 2)}$

Step 2

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$8 (x - 1) = 4 (2 x - 2)$

Step 3

Solve for $x$ .

Simplify $8 (x - 1)$ .

Start simplifying.

$0 + 0 + 8 (x - 1) = 4 (2 x - 2)$

Simplify by adding zeros.

$8 (x - 1) = 4 (2 x - 2)$

Apply the distributive property.

$8 x + 8 \cdot - 1 = 4 (2 x - 2)$

Multiply $8$ by $- 1$ .

$8 x - 8 = 4 (2 x - 2)$

Simplify $4 (2 x - 2)$ .

Apply the distributive property.

$8 x - 8 = 4 (2 x) + 4 \cdot - 2$

Multiply.

Multiply $2$ by $4$ .

$8 x - 8 = 8 x + 4 \cdot - 2$

Multiply $4$ by $- 2$ .

$8 x - 8 = 8 x - 8$

Move all terms containing $x$ to the left side of the equation.

Subtract $8 x$ from both sides of the equation.

$8 x - 8 - 8 x = - 8$

Combine the opposite terms in $8 x - 8 - 8 x$ .

Subtract $8 x$ from $8 x$ .

$0 - 8 = - 8$

Subtract $8$ from $0$ .

$- 8 = - 8$

Since $- 8 = - 8$ , the equation will always be true for any value of $x$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$