Calculus Examples

$2 \ln (2 x) + \ln (16 x) = 0$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$2 \ln (2 x) + \ln (16 x) = 0$

Step 2

Simplify the left side.

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Step 2.1

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

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

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

$\ln ({(2 x)}^{2}) + \ln (16 x) = 0$

Step 2.1.1.2

Apply the product rule to $2 x$ .

$\ln (2^{2} x^{2}) + \ln (16 x) = 0$

Step 2.1.1.3

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

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

Step 2.1.2

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

$\ln (4 x^{2} (16 x)) = 0$

Step 2.1.3

Rewrite using the commutative property of multiplication.

$\ln (4 \cdot 16 x^{2} x) = 0$

Step 2.1.4

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1.4.1

Move $x$ .

$\ln (4 \cdot 16 (x \cdot x^{2})) = 0$

Step 2.1.4.2

Multiply $x$ by $x^{2}$ .

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Step 2.1.4.2.1

Raise $x$ to the power of $1$ .

$\ln (4 \cdot 16 (x^{1} x^{2})) = 0$

Step 2.1.4.2.2

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

$\ln (4 \cdot 16 x^{1 + 2}) = 0$

Step 2.1.4.3

Add $1$ and $2$ .

$\ln (4 \cdot 16 x^{3}) = 0$

Step 2.1.5

Multiply $4$ by $16$ .

$\ln (64 x^{3}) = 0$

Step 3

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (64 x^{3})} = e^{0}$

Step 4

Rewrite $\ln (64 x^{3}) = 0$ 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 $b^{y} = x$ .

$e^{0} = 64 x^{3}$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $64 x^{3} = e^{0}$ .

$64 x^{3} = e^{0}$

Step 5.2

Subtract $e^{0}$ from both sides of the equation.

$64 x^{3} - e^{0} = 0$

Step 5.3

Simplify each term.

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Step 5.3.1

Anything raised to $0$ is $1$ .

$64 x^{3} - 1 \cdot 1 = 0$

Step 5.3.2

Multiply $- 1$ by $1$ .

$64 x^{3} - 1 = 0$

Step 5.4

Factor the left side of the equation.

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Step 5.4.1

Rewrite $64 x^{3}$ as ${(4 x)}^{3}$ .

${(4 x)}^{3} - 1 = 0$

Step 5.4.2

Rewrite $1$ as $1^{3}$ .

${(4 x)}^{3} - 1^{3} = 0$

Step 5.4.3

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 $a = 4 x$ and $b = 1$ .

$(4 x - 1) ({(4 x)}^{2} + 4 x \cdot 1 + 1^{2}) = 0$

Step 5.4.4

Simplify.

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Step 5.4.4.1

Apply the product rule to $4 x$ .

$(4 x - 1) (4^{2} x^{2} + 4 x \cdot 1 + 1^{2}) = 0$

Step 5.4.4.2

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

$(4 x - 1) (16 x^{2} + 4 x \cdot 1 + 1^{2}) = 0$

Step 5.4.4.3

Multiply $4$ by $1$ .

$(4 x - 1) (16 x^{2} + 4 x + 1^{2}) = 0$

Step 5.4.4.4

One to any power is one.

$(4 x - 1) (16 x^{2} + 4 x + 1) = 0$

Step 5.5

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

$4 x - 1 = 0$

$16 x^{2} + 4 x + 1 = 0$

Step 5.6

Set $4 x - 1$ equal to $0$ and solve for $x$ .

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Step 5.6.1

Set $4 x - 1$ equal to $0$ .

$4 x - 1 = 0$

Step 5.6.2

Solve $4 x - 1 = 0$ for $x$ .

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Step 5.6.2.1

Add $1$ to both sides of the equation.

$4 x = 1$

Step 5.6.2.2

Divide each term in $4 x = 1$ by $4$ and simplify.

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Step 5.6.2.2.1

Divide each term in $4 x = 1$ by $4$ .

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

Step 5.6.2.2.2

Simplify the left side.

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Step 5.6.2.2.2.1

Cancel the common factor of $4$ .

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Step 5.6.2.2.2.1.1

Cancel the common factor.

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

Step 5.6.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 5.7

Set $16 x^{2} + 4 x + 1$ equal to $0$ and solve for $x$ .

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Step 5.7.1

Set $16 x^{2} + 4 x + 1$ equal to $0$ .

$16 x^{2} + 4 x + 1 = 0$

Step 5.7.2

Solve $16 x^{2} + 4 x + 1 = 0$ for $x$ .

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Step 5.7.2.1

Use the quadratic formula to find the solutions.

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

Step 5.7.2.2

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

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

Step 5.7.2.3

Simplify.

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Step 5.7.2.3.1

Simplify the numerator.

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Step 5.7.2.3.1.1

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

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

Step 5.7.2.3.1.2

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

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Step 5.7.2.3.1.2.1

Multiply $- 4$ by $16$ .

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

Step 5.7.2.3.1.2.2

Multiply $- 64$ by $1$ .

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

Step 5.7.2.3.1.3

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 16}$

Step 5.7.2.3.1.4

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

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 16}$

Step 5.7.2.3.1.5

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

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 16}$

Step 5.7.2.3.1.6

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

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 16}$

Step 5.7.2.3.1.7

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

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Step 5.7.2.3.1.7.1

Factor $16$ out of $48$ .

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

Step 5.7.2.3.1.7.2

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

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

Step 5.7.2.3.1.8

Pull terms out from under the radical.

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

Step 5.7.2.3.1.9

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

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

Step 5.7.2.3.2

Multiply $2$ by $16$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{32}$

Step 5.7.2.3.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{32}$ .

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

Step 5.7.2.4

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{3}}{8}, - \frac{1 + i \sqrt{3}}{8}$

Step 5.8

The final solution is all the values that make $(4 x - 1) (16 x^{2} + 4 x + 1) = 0$ true.

$x = \frac{1}{4}, - \frac{1 - i \sqrt{3}}{8}, - \frac{1 + i \sqrt{3}}{8}$