Algebra Examples

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

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln ({(\frac{1}{2})}^{x^{2} - x - 2}) = \ln (1)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln ({(\frac{1}{2})}^{x^{2} - x - 2})$ by moving $x^{2} - x - 2$ outside the logarithm.

$(x^{2} - x - 2) \ln (\frac{1}{2}) = \ln (1)$

Step 2.2

Rewrite $\ln (\frac{1}{2})$ as $\ln (1) - \ln (2)$ .

$(x^{2} - x - 2) (\ln (1) - \ln (2)) = \ln (1)$

Step 2.3

The natural logarithm of $1$ is $0$ .

$(x^{2} - x - 2) (0 - \ln (2)) = \ln (1)$

Step 2.4

Subtract $\ln (2)$ from $0$ .

$(x^{2} - x - 2) (- \ln (2)) = \ln (1)$

Step 3

Simplify the left side.

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Step 3.1

Simplify $(x^{2} - x - 2) (- \ln (2))$ .

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Step 3.1.1

Apply the distributive property.

$x^{2} (- \ln (2)) - x (- \ln (2)) - 2 (- \ln (2)) = \ln (1)$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Multiply $- x (- \ln (2))$ .

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Step 3.1.2.1.1

Multiply $- 1$ by $- 1$ .

$x^{2} (- \ln (2)) + 1 x \ln (2) - 2 (- \ln (2)) = \ln (1)$

Step 3.1.2.1.2

Multiply $x$ by $1$ .

$x^{2} (- \ln (2)) + x \ln (2) - 2 (- \ln (2)) = \ln (1)$

Step 3.1.2.2

Multiply $- 1$ by $- 2$ .

$x^{2} (- \ln (2)) + x \ln (2) + 2 \ln (2) = \ln (1)$

Step 3.1.3

Reorder factors in $x^{2} (- \ln (2)) + x \ln (2) + 2 \ln (2)$ .

$- x^{2} \ln (2) + x \ln (2) + 2 \ln (2) = \ln (1)$

Step 4

Simplify the right side.

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Step 4.1

The natural logarithm of $1$ is $0$ .

$- x^{2} \ln (2) + x \ln (2) + 2 \ln (2) = 0$

Step 5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6

Substitute the values $a = - \ln (2)$ , $b = \ln (2)$ , and $c = 2 \ln (2)$ into the quadratic formula and solve for $x$ .

$\frac{- \ln (2) \pm \sqrt{\ln^{2} (2) - 4 \cdot (- \ln (2) \cdot (2 \ln (2)))}}{2 (- \ln (2))}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Factor $\ln (2)$ out of $\ln^{2} (2) - 4 \cdot - 1 \ln (2) \cdot (2 \ln (2))$ .

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Step 7.1.1.1

Factor $\ln (2)$ out of $\ln^{2} (2)$ .

$x = \frac{- \ln (2) \pm \sqrt{\ln (2) \ln (2) - 4 \cdot (- 1 \ln (2)) \cdot (2 \ln (2))}}{2 (- \ln (2))}$

Step 7.1.1.2

Factor $\ln (2)$ out of $- 4 \cdot - 1 \ln (2) \cdot (2 \ln (2))$ .

$x = \frac{- \ln (2) \pm \sqrt{\ln (2) \ln (2) + \ln (2) (- 4 \cdot - 1 \cdot (2 \ln (2)))}}{2 (- \ln (2))}$

Step 7.1.1.3

Factor $\ln (2)$ out of $\ln (2) \ln (2) + \ln (2) (- 4 \cdot - 1 \cdot (2 \ln (2)))$ .

$x = \frac{- \ln (2) \pm \sqrt{\ln (2) (\ln (2) - 4 \cdot - 1 \cdot (2 \ln (2)))}}{2 (- \ln (2))}$

Step 7.1.2

Factor $\ln (2)$ out of $\ln (2) - 4 \cdot - 1 \cdot 2 \ln (2)$ .

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Step 7.1.2.1

Raise $\ln (2)$ to the power of $1$ .

$x = \frac{- \ln (2) \pm \sqrt{\ln (2) (\ln (2) - 4 \cdot - 1 \cdot (2 \ln (2)))}}{2 (- \ln (2))}$

Step 7.1.2.2

Factor $\ln (2)$ out of $\ln^{1} (2)$ .

$x = \frac{- \ln (2) \pm \sqrt{\ln (2) (\ln (2) \cdot 1 - 4 \cdot - 1 \cdot (2 \ln (2)))}}{2 (- \ln (2))}$

Step 7.1.2.3

Factor $\ln (2)$ out of $- 4 \cdot - 1 \cdot 2 \ln (2)$ .

$x = \frac{- \ln (2) \pm \sqrt{\ln (2) (\ln (2) \cdot 1 + \ln (2) (- 4 \cdot - 1 \cdot 2))}}{2 (- \ln (2))}$

Step 7.1.2.4

Factor $\ln (2)$ out of $\ln (2) \cdot 1 + \ln (2) (- 4 \cdot - 1 \cdot 2)$ .

$x = \frac{- \ln (2) \pm \sqrt{\ln (2) (\ln (2) (1 - 4 \cdot - 1 \cdot 2))}}{2 (- \ln (2))}$

Step 7.1.3

Multiply $- 4 \cdot - 1 \cdot 2$ .

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Step 7.1.3.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- \ln (2) \pm \sqrt{\ln (2) (\ln (2) (1 + 4 \cdot 2))}}{2 (- \ln (2))}$

Step 7.1.3.2

Multiply $4$ by $2$ .

$x = \frac{- \ln (2) \pm \sqrt{\ln (2) (\ln (2) (1 + 8))}}{2 (- \ln (2))}$

Step 7.1.4

Add $1$ and $8$ .

$x = \frac{- \ln (2) \pm \sqrt{\ln (2) \ln (2) \cdot 9}}{2 (- \ln (2))}$

Step 7.1.5

Combine exponents.

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Step 7.1.5.1

Raise $\ln (2)$ to the power of $1$ .

$x = \frac{- \ln (2) \pm \sqrt{\ln (2) \ln (2) \cdot 9}}{2 (- \ln (2))}$

Step 7.1.5.2

Raise $\ln (2)$ to the power of $1$ .

$x = \frac{- \ln (2) \pm \sqrt{\ln (2) \ln (2) \cdot 9}}{2 (- \ln (2))}$

Step 7.1.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{- \ln (2) \pm \sqrt{{\ln (2)}^{1 + 1} \cdot 9}}{2 (- \ln (2))}$

Step 7.1.5.4

Add $1$ and $1$ .

$x = \frac{- \ln (2) \pm \sqrt{\ln^{2} (2) \cdot 9}}{2 (- \ln (2))}$

Step 7.1.6

Rewrite $\ln^{2} (2) \cdot 9$ as ${(\ln (2) \cdot 3)}^{2}$ .

$x = \frac{- \ln (2) \pm \sqrt{{(\ln (2) \cdot 3)}^{2}}}{2 (- \ln (2))}$

Step 7.1.7

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- \ln (2) \pm \ln (2) \cdot 3}{2 (- \ln (2))}$

Step 7.1.8

Move $3$ to the left of $\ln (2)$ .

$x = \frac{- \ln (2) \pm 3 \ln (2)}{2 (- \ln (2))}$

Step 7.2

Multiply $- 1$ by $2$ .

$x = \frac{- \ln (2) \pm 3 \ln (2)}{- 2 \ln (2)}$

Step 7.3

Simplify $\frac{- \ln (2) \pm 3 \ln (2)}{- 2 \ln (2)}$ .

$x = \frac{1 \pm 3}{2}$

Step 8

The final answer is the combination of both solutions.

$x = 2, - 1$