Calculus Examples

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

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

Step 1

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

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

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

Step 2

Simplify the left side.

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Step 2.1

Simplify

2 ln (2 x) + ln (16 x)

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

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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)

$2 \ln (2 x)$ by moving

2

$2$ inside the logarithm.

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

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

Step 2.1.1.2

Apply the product rule to

2 x

$2 x$ .

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

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

Step 2.1.1.3

Raise

2

$2$ to the power of

2

$2$ .

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

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

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

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

Step 2.1.2

Use the product property of logarithms,

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

$\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

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

Step 2.1.3

Rewrite using the commutative property of multiplication.

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

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

Step 2.1.4

Multiply

x^{2}

$x^{2}$ by

x

$x$ by adding the exponents.

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Step 2.1.4.1

Move

x

$x$ .

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

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

Step 2.1.4.2

Multiply

x

$x$ by

x^{2}

$x^{2}$ .

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Step 2.1.4.2.1

Raise

x

$x$ to the power of

1

$1$ .

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

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

Step 2.1.4.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

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

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

Step 2.1.4.3

Add

1

$1$ and

2

$2$ .

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

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

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

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

Step 2.1.5

Multiply

4

$4$ by

16

$16$ .

ln (64 x^{3}) = 0

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

ln (64 x^{3}) = 0

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

ln (64 x^{3}) = 0

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

Step 3

To solve for

x

$x$ , rewrite the equation using properties of logarithms.

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

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

Step 4

Rewrite

ln (64 x^{3}) = 0

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

$b^{y} = x$ .

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

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

Step 5

Solve for

x

$x$ .

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Step 5.1

Rewrite the equation as

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

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

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

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

Step 5.2

Subtract

e^{0}

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

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

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

Step 5.3

Simplify each term.

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Step 5.3.1

Anything raised to

0

$0$ is

1

$1$ .

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

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

Step 5.3.2

Multiply

- 1

$- 1$ by

1

$1$ .

64 x^{3} - 1 = 0

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

64 x^{3} - 1 = 0

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

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.

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