Calculus Examples

$\ln (x) = \frac{1}{3} \cdot (\ln (16) + 2 \ln (2))$

Step 1

Simplify the right side.

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Step 1.1

Simplify $\frac{1}{3} \cdot (\ln (16) + 2 \ln (2))$ .

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Step 1.1.1

Apply the distributive property.

$\ln (x) = \frac{1}{3} \ln (16) + \frac{1}{3} (2 \ln (2))$

Step 1.1.2

Combine $\frac{1}{3}$ and $\ln (16)$ .

$\ln (x) = \frac{\ln (16)}{3} + \frac{1}{3} (2 \ln (2))$

Step 1.1.3

Multiply $\frac{1}{3} (2 \ln (2))$ .

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Step 1.1.3.1

Combine $2$ and $\frac{1}{3}$ .

$\ln (x) = \frac{\ln (16)}{3} + \frac{2}{3} \ln (2)$

Step 1.1.3.2

Combine $\frac{2}{3}$ and $\ln (2)$ .

$\ln (x) = \frac{\ln (16)}{3} + \frac{2 \ln (2)}{3}$

Step 2

Multiply each term in $\ln (x) = \frac{\ln (16)}{3} + \frac{2 \ln (2)}{3}$ by $3$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\ln (x) = \frac{\ln (16)}{3} + \frac{2 \ln (2)}{3}$ by $3$ .

$\ln (x) \cdot 3 = \frac{\ln (16)}{3} \cdot 3 + \frac{2 \ln (2)}{3} \cdot 3$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

$3 \ln (x) = \frac{\ln (16)}{3} \cdot 3 + \frac{2 \ln (2)}{3} \cdot 3$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Cancel the common factor of $3$ .

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Step 2.3.1.1.1

Cancel the common factor.

$3 \ln (x) = \frac{\ln (16)}{3} \cdot 3 + \frac{2 \ln (2)}{3} \cdot 3$

Step 2.3.1.1.2

Rewrite the expression.

$3 \ln (x) = \ln (16) + \frac{2 \ln (2)}{3} \cdot 3$

Step 2.3.1.2

Cancel the common factor of $3$ .

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Step 2.3.1.2.1

Cancel the common factor.

$3 \ln (x) = \ln (16) + \frac{2 \ln (2)}{3} \cdot 3$

Step 2.3.1.2.2

Rewrite the expression.

$3 \ln (x) = \ln (16) + 2 \ln (2)$

Step 3

Simplify the left side.

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Step 3.1

Simplify $3 \ln (x)$ by moving $3$ inside the logarithm.

$\ln (x^{3}) = \ln (16) + 2 \ln (2)$

Step 4

Simplify the right side.

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Step 4.1

Simplify $\ln (16) + 2 \ln (2)$ .

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Simplify $2 \ln (2)$ by moving $2$ inside the logarithm.

$\ln (x^{3}) = \ln (16) + \ln (2^{2})$

Step 4.1.1.2

Raise $2$ to the power of $2$ .

$\ln (x^{3}) = \ln (16) + \ln (4)$

Step 4.1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln (x^{3}) = \ln (16 \cdot 4)$

Step 4.1.3

Multiply $16$ by $4$ .

$\ln (x^{3}) = \ln (64)$

Step 5

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$x^{3} = 64$

Step 6

Solve for $x$ .

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Step 6.1

Subtract $64$ from both sides of the equation.

$x^{3} - 64 = 0$

Step 6.2

Factor the left side of the equation.

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Step 6.2.1

Rewrite $64$ as $4^{3}$ .

$x^{3} - 4^{3} = 0$

Step 6.2.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 4$ .

$(x - 4) (x^{2} + x \cdot 4 + 4^{2}) = 0$

Step 6.2.3

Simplify.

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Step 6.2.3.1

Move $4$ to the left of $x$ .

$(x - 4) (x^{2} + 4 x + 4^{2}) = 0$

Step 6.2.3.2

Raise $4$ to the power of $2$ .

$(x - 4) (x^{2} + 4 x + 16) = 0$

Step 6.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 4 = 0$

$x^{2} + 4 x + 16 = 0$

Step 6.4

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 6.4.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 6.4.2

Add $4$ to both sides of the equation.

$x = 4$

Step 6.5

Set $x^{2} + 4 x + 16$ equal to $0$ and solve for $x$ .

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Step 6.5.1

Set $x^{2} + 4 x + 16$ equal to $0$ .

$x^{2} + 4 x + 16 = 0$

Step 6.5.2

Solve $x^{2} + 4 x + 16 = 0$ for $x$ .

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Step 6.5.2.1

Use the quadratic formula to find the solutions.

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

Step 6.5.2.2

Substitute the values $a = 1$ , $b = 4$ , and $c = 16$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot 16)}}{2 \cdot 1}$

Step 6.5.2.3

Simplify.

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Step 6.5.2.3.1

Simplify the numerator.

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Step 6.5.2.3.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Step 6.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

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Step 6.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 16}}{2 \cdot 1}$

Step 6.5.2.3.1.2.2

Multiply $- 4$ by $16$ .

$x = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Step 6.5.2.3.1.3

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Step 6.5.2.3.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Step 6.5.2.3.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Step 6.5.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Step 6.5.2.3.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 6.5.2.3.1.7.1

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Step 6.5.2.3.1.7.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 6.5.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Step 6.5.2.3.1.9

Move $4$ to the left of $i$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Step 6.5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2}$

Step 6.5.2.3.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

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

Step 6.5.2.4

The final answer is the combination of both solutions.

$x = - 2 + 2 i \sqrt{3}, - 2 - 2 i \sqrt{3}$

Step 6.6

The final solution is all the values that make $(x - 4) (x^{2} + 4 x + 16) = 0$ true.

$x = 4, - 2 + 2 i \sqrt{3}, - 2 - 2 i \sqrt{3}$