Precalculus Examples

$9^{4 x - 1} = 27$

Step 1

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

$3^{2 (4 x - 1)} = 3^{3}$

Step 2

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

$2 (4 x - 1) = 3$

Step 3

Solve for $x$ .

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Step 3.1

Divide each term in $2 (4 x - 1) = 3$ by $2$ and simplify.

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Step 3.1.1

Divide each term in $2 (4 x - 1) = 3$ by $2$ .

$\frac{2 (4 x - 1)}{2} = \frac{3}{2}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $2$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{2 (4 x - 1)}{2} = \frac{3}{2}$

Step 3.1.2.1.2

Divide $4 x - 1$ by $1$ .

$4 x - 1 = \frac{3}{2}$

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 $1$ to both sides of the equation.

$4 x = \frac{3}{2} + 1$

Step 3.2.2

Write $1$ as a fraction with a common denominator.

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

Step 3.2.3

Combine the numerators over the common denominator.

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

Step 3.2.4

Add $3$ and $2$ .

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

Step 3.3

Divide each term in $4 x = \frac{5}{2}$ by $4$ and simplify.

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Step 3.3.1

Divide each term in $4 x = \frac{5}{2}$ by $4$ .

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $4$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Divide $x$ by $1$ .

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

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5}{2} \cdot \frac{1}{4}$

Step 3.3.3.2

Multiply $\frac{5}{2} \cdot \frac{1}{4}$ .

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Step 3.3.3.2.1

Multiply $\frac{5}{2}$ by $\frac{1}{4}$ .

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

Step 3.3.3.2.2

Multiply $2$ by $4$ .

$x = \frac{5}{8}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{8}$

Decimal Form:

$x = 0.625$