Precalculus Examples

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

Step 1

Rewrite $\log_{2} (x - 3) = - 2$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

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

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $x - 3 = 2^{- 2}$ .

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

Step 2.2

Simplify $2^{- 2}$ .

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Step 2.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.2.2

Raise $2$ to the power of $2$ .

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

Step 2.3

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

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Step 2.3.1

Add $3$ to both sides of the equation.

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

Step 2.3.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 2.3.3

Combine $3$ and $\frac{4}{4}$ .

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

Step 2.3.4

Combine the numerators over the common denominator.

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

Step 2.3.5

Simplify the numerator.

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Step 2.3.5.1

Multiply $3$ by $4$ .

$x = \frac{1 + 12}{4}$

Step 2.3.5.2

Add $1$ and $12$ .

$x = \frac{13}{4}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{13}{4}$

Decimal Form:

$x = 3.25$

Mixed Number Form:

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