Solve for z 25^(z-4)=125

$25^{z - 4} = 125$

Step 1

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

$5^{2 (z - 4)} = 5^{3}$

Step 2

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

$2 (z - 4) = 3$

Step 3

Solve for $z$ .

Divide each term in $2 (z - 4) = 3$ by $2$ and simplify.

Divide each term in $2 (z - 4) = 3$ by $2$ .

$\frac{2 (z - 4)}{2} = \frac{3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 (z - 4)}{2} = \frac{3}{2}$

Divide $z - 4$ by $1$ .

$z - 4 = \frac{3}{2}$

Move all terms not containing $z$ to the right side of the equation.

Add $4$ to both sides of the equation.

$z = \frac{3}{2} + 4$

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$z = \frac{3}{2} + 4 \cdot \frac{2}{2}$

Combine $4$ and $\frac{2}{2}$ .

$z = \frac{3}{2} + \frac{4 \cdot 2}{2}$

Combine the numerators over the common denominator.

$z = \frac{3 + 4 \cdot 2}{2}$

Simplify the numerator.

Multiply $4$ by $2$ .

$z = \frac{3 + 8}{2}$

Add $3$ and $8$ .

$z = \frac{11}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$z = \frac{11}{2}$

Decimal Form:

$z = 5.5$

Mixed Number Form:

$z = 5 \frac{1}{2}$