Precalculus Examples

$125^{x - 2} = 25$

Step 1

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

$5^{3 (x - 2)} = 5^{2}$

Step 2

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

$3 (x - 2) = 2$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

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

$\frac{3 (x - 2)}{3} = \frac{2}{3}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $3$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{3 (x - 2)}{3} = \frac{2}{3}$

Step 3.1.2.1.2

Divide $x - 2$ by $1$ .

$x - 2 = \frac{2}{3}$

Step 3.2

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

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Step 3.2.1

Add $2$ to both sides of the equation.

$x = \frac{2}{3} + 2$

Step 3.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{2}{3} + 2 \cdot \frac{3}{3}$

Step 3.2.3

Combine $2$ and $\frac{3}{3}$ .

$x = \frac{2}{3} + \frac{2 \cdot 3}{3}$

Step 3.2.4

Combine the numerators over the common denominator.

$x = \frac{2 + 2 \cdot 3}{3}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $2$ by $3$ .

$x = \frac{2 + 6}{3}$

Step 3.2.5.2

Add $2$ and $6$ .

$x = \frac{8}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{8}{3}$

Decimal Form:

$x = 2.\overline{6}$

Mixed Number Form:

$x = 2 \frac{2}{3}$