Precalculus Examples

$\frac{5}{2^{x}} = \frac{5}{4}$

Step 1

Multiply both sides by $2^{x}$ .

$\frac{5}{2^{x}} \cdot 2^{x} = \frac{5}{4} \cdot 2^{x}$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $2^{x}$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{5}{2^{x}} \cdot 2^{x} = \frac{5}{4} \cdot 2^{x}$

Step 2.1.1.2

Rewrite the expression.

$5 = \frac{5}{4} \cdot 2^{x}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Combine $\frac{5}{4}$ and $2^{x}$ .

$5 = \frac{5 \cdot 2^{x}}{4}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $\frac{5 \cdot 2^{x}}{4} = 5$ .

$\frac{5 \cdot 2^{x}}{4} = 5$

Step 3.2

Multiply both sides of the equation by $\frac{4}{5}$ .

$\frac{4}{5} \cdot \frac{5 \cdot 2^{x}}{4} = \frac{4}{5} \cdot 5$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $\frac{4}{5} \cdot \frac{5 \cdot 2^{x}}{4}$ .

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Step 3.3.1.1.1

Combine.

$\frac{4 (5 \cdot 2^{x})}{5 \cdot 4} = \frac{4}{5} \cdot 5$

Step 3.3.1.1.2

Cancel the common factor of $4$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

$\frac{4 (5 \cdot 2^{x})}{5 \cdot 4} = \frac{4}{5} \cdot 5$

Step 3.3.1.1.2.2

Rewrite the expression.

$\frac{5 \cdot 2^{x}}{5} = \frac{4}{5} \cdot 5$

Step 3.3.1.1.3

Cancel the common factor of $5$ .

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Step 3.3.1.1.3.1

Cancel the common factor.

$\frac{5 \cdot 2^{x}}{5} = \frac{4}{5} \cdot 5$

Step 3.3.1.1.3.2

Divide $2^{x}$ by $1$ .

$2^{x} = \frac{4}{5} \cdot 5$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Cancel the common factor of $5$ .

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Step 3.3.2.1.1

Cancel the common factor.

$2^{x} = \frac{4}{5} \cdot 5$

Step 3.3.2.1.2

Rewrite the expression.

$2^{x} = 4$

Step 3.4

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

$2^{x} = 2^{2}$

Step 3.5

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

$x = 2$