Precalculus Examples

$\log_{4} (x - 5) = - 1$

Step 1

Rewrite $\log_{4} (x - 5) = - 1$ 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 $b^{y} = x$ .

$4^{- 1} = x - 5$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $x - 5 = 4^{- 1}$ .

$x - 5 = 4^{- 1}$

Step 2.2

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

$x - 5 = \frac{1}{4}$

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 $5$ to both sides of the equation.

$x = \frac{1}{4} + 5$

Step 2.3.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 2.3.3

Combine $5$ and $\frac{4}{4}$ .

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

Step 2.3.4

Combine the numerators over the common denominator.

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

Step 2.3.5

Simplify the numerator.

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Step 2.3.5.1

Multiply $5$ by $4$ .

$x = \frac{1 + 20}{4}$

Step 2.3.5.2

Add $1$ and $20$ .

$x = \frac{21}{4}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{21}{4}$

Decimal Form:

$x = 5.25$

Mixed Number Form:

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