Precalculus Examples

125 = 25^{x - 1}

$125 = 25^{x - 1}$

Step 1

Rewrite the equation as

25^{x - 1} = 125

$25^{x - 1} = 125$ .

25^{x - 1} = 125

$25^{x - 1} = 125$

Step 2

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

5^{2 (x - 1)} = 5^{3}

$5^{2 (x - 1)} = 5^{3}$

Step 3

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

2 (x - 1) = 3

$2 (x - 1) = 3$

Step 4

Solve for

x

$x$ .

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Step 4.1

Divide each term in

2 (x - 1) = 3

$2 (x - 1) = 3$ by

2

$2$ and simplify.

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Step 4.1.1

Divide each term in

2 (x - 1) = 3

$2 (x - 1) = 3$ by

2

$2$ .

\frac{2 (x - 1)}{2} = \frac{3}{2}

$\frac{2 (x - 1)}{2} = \frac{3}{2}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of

2

$2$ .

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Step 4.1.2.1.1

Cancel the common factor.

\frac{2 (x - 1)}{2} = \frac{3}{2}

$\frac{2 (x - 1)}{2} = \frac{3}{2}$

Step 4.1.2.1.2

Divide

x - 1

$x - 1$ by

1

$1$ .

x - 1 = \frac{3}{2}

$x - 1 = \frac{3}{2}$

x - 1 = \frac{3}{2}

$x - 1 = \frac{3}{2}$

x - 1 = \frac{3}{2}

$x - 1 = \frac{3}{2}$

x - 1 = \frac{3}{2}

$x - 1 = \frac{3}{2}$

Step 4.2

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 4.2.1

Add

1

$1$ to both sides of the equation.

x = \frac{3}{2} + 1

$x = \frac{3}{2} + 1$

Step 4.2.2

Write

1

$1$ as a fraction with a common denominator.

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

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

Step 4.2.3

Combine the numerators over the common denominator.

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

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

Step 4.2.4

Add

3

$3$ and

2

$2$ .

x = \frac{5}{2}

$x = \frac{5}{2}$

x = \frac{5}{2}

$x = \frac{5}{2}$

x = \frac{5}{2}

$x = \frac{5}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

x = \frac{5}{2}

$x = \frac{5}{2}$

Decimal Form:

x = 2.5

$x = 2.5$

Mixed Number Form:

x = 2 \frac{1}{2}

$x = 2 \frac{1}{2}$