Precalculus Examples

$\log_{3} (x + 4) = - 3$

Step 1

Rewrite $\log_{3} (x + 4) = - 3$ 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 $b^{y} = x$ .

$3^{- 3} = x + 4$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $x + 4 = 3^{- 3}$ .

$x + 4 = 3^{- 3}$

Step 2.2

Simplify $3^{- 3}$ .

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Step 2.2.1

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

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

Step 2.2.2

Raise $3$ to the power of $3$ .

$x + 4 = \frac{1}{27}$

Step 2.3

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

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Step 2.3.1

Subtract $4$ from both sides of the equation.

$x = \frac{1}{27} - 4$

Step 2.3.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$x = \frac{1}{27} - 4 \cdot \frac{27}{27}$

Step 2.3.3

Combine $- 4$ and $\frac{27}{27}$ .

$x = \frac{1}{27} + \frac{- 4 \cdot 27}{27}$

Step 2.3.4

Combine the numerators over the common denominator.

$x = \frac{1 - 4 \cdot 27}{27}$

Step 2.3.5

Simplify the numerator.

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Step 2.3.5.1

Multiply $- 4$ by $27$ .

$x = \frac{1 - 108}{27}$

Step 2.3.5.2

Subtract $108$ from $1$ .

$x = \frac{- 107}{27}$

Step 2.3.6

Move the negative in front of the fraction.

$x = - \frac{107}{27}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{107}{27}$

Decimal Form:

$x = - 3.\overline{962}$

Mixed Number Form:

$x = - 3 \frac{26}{27}$