Precalculus Examples

$\frac{1}{27} = 3^{2 x}$

Step 1

Rewrite the equation as $3^{2 x} = \frac{1}{27}$ .

$3^{2 x} = \frac{1}{27}$

Step 2

Raise $27$ to the power of $1$ .

$3^{2 x} = \frac{1}{27^{1}}$

Step 3

Move $27^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$3^{2 x} = 27^{- 1}$

Step 4

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

$3^{2 x} = 3^{- 3}$

Step 5

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

$2 x = - 3$

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Divide $x$ by $1$ .

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

Step 6.3

Simplify the right side.

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Step 6.3.1

Move the negative in front of the fraction.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 1.5$

Mixed Number Form:

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