Precalculus Examples

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

Step 1

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

$\ln (2^{x^{3} - x}) = \ln (1)$

Step 2

Expand

$\ln (2^{x^{3} - x})$ by moving

$x^{3} - x$ outside the logarithm.

$(x^{3} - x) \ln (2) = \ln (1)$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$x^{3} \ln (2) - x \ln (2) = \ln (1)$

Step 4

Simplify the right side.

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Step 4.1

The natural logarithm of

$1$ is

$0$ .

$x^{3} \ln (2) - x \ln (2) = 0$

Step 5

Factor

$x \ln (2)$ out of

$x^{3} \ln (2) - x \ln (2)$ .

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Step 5.1

Factor

$x \ln (2)$ out of

$x^{3} \ln (2)$ .

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

Step 5.2

Factor

$x \ln (2)$ out of

$- x \ln (2)$ .

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

Step 5.3

Factor

$x \ln (2)$ out of

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

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

Step 6

Rewrite

$1$ as

$1^{2}$ .

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

Step 7

Factor.

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Step 7.1

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = x$ and

$b = 1$ .

$x \ln (2) ((x + 1) (x - 1)) = 0$

Step 7.2

Remove unnecessary parentheses.

$x \ln (2) (x + 1) (x - 1) = 0$

Step 8

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$x = 0$

$x + 1 = 0$

$x - 1 = 0$

Step 9

Set

$x$ equal to

$0$ .

$x = 0$

Step 10

Set

$x + 1$ equal to

$0$ and solve for

$x$ .

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Step 10.1

Set

$x + 1$ equal to

$0$ .

$x + 1 = 0$

Step 10.2

Subtract

$1$ from both sides of the equation.

$x = - 1$

Step 11

Set

$x - 1$ equal to

$0$ and solve for

$x$ .

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Step 11.1

Set

$x - 1$ equal to

$0$ .

$x - 1 = 0$

Step 11.2

Add

$1$ to both sides of the equation.

$x = 1$

Step 12

The final solution is all the values that make

$x \ln (2) (x + 1) (x - 1) = 0$ true.

$x = 0, - 1, 1$