Precalculus Examples

3 = {log}_{x} (512)

$3 = \log_{x} (512)$

Step 1

Rewrite the equation as

{log}_{x} (512) = 3

$\log_{x} (512) = 3$ .

{log}_{x} (512) = 3

$\log_{x} (512) = 3$

Step 2

Rewrite

{log}_{x} (512) = 3

$\log_{x} (512) = 3$ in exponential form using the definition of a logarithm. If

x

$x$ and

b

$b$ are positive real numbers and

b \neq 1

$b \neq 1$ , then

{log}_{b} (x) = y

$\log_{b} (x) = y$ is equivalent to

b^{y} = x

$b^{y} = x$ .

x^{3} = 512

$x^{3} = 512$

Step 3

Solve for

x

$x$ .

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Step 3.1

Subtract

512

$512$ from both sides of the equation.

x^{3} - 512 = 0

$x^{3} - 512 = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Rewrite

512

$512$ as

8^{3}

$8^{3}$ .

x^{3} - 8^{3} = 0

$x^{3} - 8^{3} = 0$

Step 3.2.2

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

a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})

$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where

a = x

$a = x$ and

b = 8

$b = 8$ .

(x - 8) (x^{2} + x \cdot 8 + 8^{2}) = 0

$(x - 8) (x^{2} + x \cdot 8 + 8^{2}) = 0$

Step 3.2.3

Simplify.

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Step 3.2.3.1

Move

8

$8$ to the left of

x

$x$ .

(x - 8) (x^{2} + 8 x + 8^{2}) = 0

$(x - 8) (x^{2} + 8 x + 8^{2}) = 0$

Step 3.2.3.2

Raise

8

$8$ to the power of

2

$2$ .

(x - 8) (x^{2} + 8 x + 64) = 0

$(x - 8) (x^{2} + 8 x + 64) = 0$

(x - 8) (x^{2} + 8 x + 64) = 0

$(x - 8) (x^{2} + 8 x + 64) = 0$

(x - 8) (x^{2} + 8 x + 64) = 0

$(x - 8) (x^{2} + 8 x + 64) = 0$

Step 3.3

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

0

$0$ , the entire expression will be equal to

0

$0$ .

x - 8 = 0

$x - 8 = 0$

x^{2} + 8 x + 64 = 0

$x^{2} + 8 x + 64 = 0$

Step 3.4

Set

x - 8

$x - 8$ equal to

0

$0$ and solve for

x

$x$ .

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Step 3.4.1

Set

x - 8

$x - 8$ equal to

0

$0$ .

x - 8 = 0

$x - 8 = 0$

Step 3.4.2

Add

8

$8$ to both sides of the equation.

x = 8

$x = 8$

x = 8

$x = 8$

Step 3.5

Set

x^{2} + 8 x + 64

$x^{2} + 8 x + 64$ equal to

0

$0$ and solve for

x

$x$ .

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Step 3.5.1

Set

x^{2} + 8 x + 64

$x^{2} + 8 x + 64$ equal to

0

$0$ .

x^{2} + 8 x + 64 = 0

$x^{2} + 8 x + 64 = 0$

Step 3.5.2

Solve

x^{2} + 8 x + 64 = 0

$x^{2} + 8 x + 64 = 0$ for

x

$x$ .

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Step 3.5.2.1

Use the quadratic formula to find the solutions.

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

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

Step 3.5.2.2

Substitute the values

a = 1

$a = 1$ ,

b = 8

$b = 8$ , and

c = 64

$c = 64$ into the quadratic formula and solve for

x

$x$ .

\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (1 \cdot 64)}}{2 \cdot 1}

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (1 \cdot 64)}}{2 \cdot 1}$

Step 3.5.2.3

Simplify.

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Step 3.5.2.3.1

Simplify the numerator.

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Step 3.5.2.3.1.1

Raise

8

$8$ to the power of

2

$2$ .

x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 1 \cdot 64}}{2 \cdot 1}

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 1 \cdot 64}}{2 \cdot 1}$

Step 3.5.2.3.1.2

Multiply

- 4 \cdot 1 \cdot 64

$- 4 \cdot 1 \cdot 64$ .

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Step 3.5.2.3.1.2.1

Multiply

- 4

$- 4$ by

1

$1$ .

x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 64}}{2 \cdot 1}

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 64}}{2 \cdot 1}$

Step 3.5.2.3.1.2.2

Multiply

- 4

$- 4$ by

64

$64$ .

x = \frac{- 8 \pm \sqrt{64 - 256}}{2 \cdot 1}

$x = \frac{- 8 \pm \sqrt{64 - 256}}{2 \cdot 1}$

x = \frac{- 8 \pm \sqrt{64 - 256}}{2 \cdot 1}

$x = \frac{- 8 \pm \sqrt{64 - 256}}{2 \cdot 1}$

Step 3.5.2.3.1.3

Subtract

256

$256$ from

64

$64$ .

x = \frac{- 8 \pm \sqrt{- 192}}{2 \cdot 1}

$x = \frac{- 8 \pm \sqrt{- 192}}{2 \cdot 1}$

Step 3.5.2.3.1.4

Rewrite

- 192

$- 192$ as

- 1 (192)

$- 1 (192)$ .

x = \frac{- 8 \pm \sqrt{- 1 \cdot 192}}{2 \cdot 1}

$x = \frac{- 8 \pm \sqrt{- 1 \cdot 192}}{2 \cdot 1}$

Step 3.5.2.3.1.5

Rewrite

\sqrt{- 1 (192)}

$\sqrt{- 1 (192)}$ as

\sqrt{- 1} \cdot \sqrt{192}

$\sqrt{- 1} \cdot \sqrt{192}$ .

x = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{192}}{2 \cdot 1}

$x = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{192}}{2 \cdot 1}$

Step 3.5.2.3.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

$x = \frac{- 8 \pm i \cdot \sqrt{192}}{2 \cdot 1}$

Step 3.5.2.3.1.7

Rewrite

$192$ as

$8^{2} \cdot 3$ .

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Step 3.5.2.3.1.7.1

Factor

$64$ out of

$192$ .

$x = \frac{- 8 \pm i \cdot \sqrt{64 (3)}}{2 \cdot 1}$

Step 3.5.2.3.1.7.2

Rewrite

$64$ as

$8^{2}$ .

$x = \frac{- 8 \pm i \cdot \sqrt{8^{2} \cdot 3}}{2 \cdot 1}$

Step 3.5.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 8 \pm i \cdot (8 \sqrt{3})}{2 \cdot 1}$

Step 3.5.2.3.1.9

Move

$8$ to the left of

$i$ .

$x = \frac{- 8 \pm 8 i \sqrt{3}}{2 \cdot 1}$

Step 3.5.2.3.2

Multiply

$2$ by

$1$ .

$x = \frac{- 8 \pm 8 i \sqrt{3}}{2}$

Step 3.5.2.3.3

Simplify

$\frac{- 8 \pm 8 i \sqrt{3}}{2}$ .

$x = - 4 \pm 4 i \sqrt{3}$

Step 3.5.2.4

The final answer is the combination of both solutions.

$x = - 4 + 4 i \sqrt{3}, - 4 - 4 i \sqrt{3}$

Step 3.6

The final solution is all the values that make

$(x - 8) (x^{2} + 8 x + 64) = 0$ true.

$x = 8, - 4 + 4 i \sqrt{3}, - 4 - 4 i \sqrt{3}$