Precalculus Examples

$e^{4 x} - 6 e^{2 x} = 16$

Step 1

Rewrite $e^{4 x}$ as exponentiation.

${(e^{x})}^{4} - 6 e^{2 x} = 16$

Step 2

Rewrite $e^{2 x}$ as exponentiation.

${(e^{x})}^{4} - 6 {(e^{x})}^{2} = 16$

Step 3

Substitute $u$ for $e^{x}$ .

$u^{4} - 6 u^{2} = 16$

Step 4

Solve for $u$ .

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Step 4.1

Subtract $16$ from both sides of the equation.

$u^{4} - 6 u^{2} - 16 = 0$

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

Rewrite $u^{4}$ as ${(u^{2})}^{2}$ .

${(u^{2})}^{2} - 6 u^{2} - 16 = 0$

Step 4.2.2

Let $u = u^{2}$ . Substitute $u$ for all occurrences of $u^{2}$ .

$u^{2} - 6 u - 16 = 0$

Step 4.2.3

Factor $u^{2} - 6 u - 16$ using the AC method.

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Step 4.2.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 16$ and whose sum is $- 6$ .

$- 8, 2$

Step 4.2.3.2

Write the factored form using these integers.

$(u - 8) (u + 2) = 0$

Step 4.2.4

Replace all occurrences of $u$ with $u^{2}$ .

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

Step 4.3

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

$u^{2} - 8 = 0$

$u^{2} + 2 = 0$

Step 4.4

Set $u^{2} - 8$ equal to $0$ and solve for $u$ .

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Step 4.4.1

Set $u^{2} - 8$ equal to $0$ .

$u^{2} - 8 = 0$

Step 4.4.2

Solve $u^{2} - 8 = 0$ for $u$ .

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Step 4.4.2.1

Add $8$ to both sides of the equation.

$u^{2} = 8$

Step 4.4.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$u = \pm \sqrt{8}$

Step 4.4.2.3

Simplify $\pm \sqrt{8}$ .

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Step 4.4.2.3.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.4.2.3.1.1

Factor $4$ out of $8$ .

$u = \pm \sqrt{4 (2)}$

Step 4.4.2.3.1.2

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

$u = \pm \sqrt{2^{2} \cdot 2}$

Step 4.4.2.3.2

Pull terms out from under the radical.

$u = \pm 2 \sqrt{2}$

Step 4.4.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.4.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$u = 2 \sqrt{2}$

Step 4.4.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$u = - 2 \sqrt{2}$

Step 4.4.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$u = 2 \sqrt{2}, - 2 \sqrt{2}$

Step 4.5

Set $u^{2} + 2$ equal to $0$ and solve for $u$ .

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Step 4.5.1

Set $u^{2} + 2$ equal to $0$ .

$u^{2} + 2 = 0$

Step 4.5.2

Solve $u^{2} + 2 = 0$ for $u$ .

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Step 4.5.2.1

Subtract $2$ from both sides of the equation.

$u^{2} = - 2$

Step 4.5.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$u = \pm \sqrt{- 2}$

Step 4.5.2.3

Simplify $\pm \sqrt{- 2}$ .

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Step 4.5.2.3.1

Rewrite $- 2$ as $- 1 (2)$ .

$u = \pm \sqrt{- 1 (2)}$

Step 4.5.2.3.2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$u = \pm \sqrt{- 1} \cdot \sqrt{2}$

Step 4.5.2.3.3

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

$u = \pm i \sqrt{2}$

Step 4.5.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.5.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$u = i \sqrt{2}$

Step 4.5.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$u = - i \sqrt{2}$

Step 4.5.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$u = i \sqrt{2}, - i \sqrt{2}$

Step 4.6

The final solution is all the values that make $(u^{2} - 8) (u^{2} + 2) = 0$ true.

$u = 2 \sqrt{2}, - 2 \sqrt{2}, i \sqrt{2}, - i \sqrt{2}$

Step 5

Substitute $2 \sqrt{2}$ for $u$ in $u = e^{x}$ .

$2 \sqrt{2} = e^{x}$

Step 6

Solve $2 \sqrt{2} = e^{x}$ .

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Step 6.1

Rewrite the equation as $e^{x} = 2 \sqrt{2}$ .

$e^{x} = 2 \sqrt{2}$

Step 6.2

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

$\ln (e^{x}) = \ln (2 \sqrt{2})$

Step 6.3

Expand the left side.

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Step 6.3.1

Expand $\ln (e^{x})$ by moving $x$ outside the logarithm.

$x \ln (e) = \ln (2 \sqrt{2})$

Step 6.3.2

The natural logarithm of $e$ is $1$ .

$x \cdot 1 = \ln (2 \sqrt{2})$

Step 6.3.3

Multiply $x$ by $1$ .

$x = \ln (2 \sqrt{2})$

Step 7

Substitute $- 2 \sqrt{2}$ for $u$ in $u = e^{x}$ .

$- 2 \sqrt{2} = e^{x}$

Step 8

Solve $- 2 \sqrt{2} = e^{x}$ .

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Step 8.1

Rewrite the equation as $e^{x} = - 2 \sqrt{2}$ .

$e^{x} = - 2 \sqrt{2}$

Step 8.2

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

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

Step 8.3

The equation cannot be solved because $\ln (- 2 \sqrt{2})$ is undefined.

Undefined

Step 8.4

There is no solution for $e^{x} = - 2 \sqrt{2}$

No solution

Step 9

Substitute $i \sqrt{2}$ for $u$ in $u = e^{x}$ .

$i \sqrt{2} = e^{x}$

Step 10

Solve $i \sqrt{2} = e^{x}$ .

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Step 10.1

Rewrite the equation as $e^{x} = i \sqrt{2}$ .

$e^{x} = i \sqrt{2}$

Step 10.2

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

$\ln (e^{x}) = \ln (i \sqrt{2})$

Step 10.3

Expand the left side.

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Step 10.3.1

Expand $\ln (e^{x})$ by moving $x$ outside the logarithm.

$x \ln (e) = \ln (i \sqrt{2})$

Step 10.3.2

The natural logarithm of $e$ is $1$ .

$x \cdot 1 = \ln (i \sqrt{2})$

Step 10.3.3

Multiply $x$ by $1$ .

$x = \ln (i \sqrt{2})$

Step 10.4

Expand the right side.

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Step 10.4.1

Rewrite $\ln (i \sqrt{2})$ as $\ln (i) + \ln (\sqrt{2})$ .

$x = \ln (i) + \ln (\sqrt{2})$

Step 10.4.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \ln (i) + \ln (2^{\frac{1}{2}})$

Step 10.4.3

Expand $\ln (2^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 10.4.4

Combine $\frac{1}{2}$ and $\ln (2)$ .

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

Step 10.5

Simplify.

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Step 10.5.1

Simplify each term.

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Step 10.5.1.1

Rewrite $\frac{\ln (2)}{2}$ as $\frac{1}{2} \ln (2)$ .

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

Step 10.5.1.2

Simplify $\frac{1}{2} \ln (2)$ by moving $\frac{1}{2}$ inside the logarithm.

$x = \ln (i) + \ln (2^{\frac{1}{2}})$

Step 10.5.2

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

$x = \ln (i \cdot 2^{\frac{1}{2}})$

Step 11

Substitute $- i \sqrt{2}$ for $u$ in $u = e^{x}$ .

$- i \sqrt{2} = e^{x}$

Step 12

Solve $- i \sqrt{2} = e^{x}$ .

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Step 12.1

Rewrite the equation as $e^{x} = - i \sqrt{2}$ .

$e^{x} = - i \sqrt{2}$

Step 12.2

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

$\ln (e^{x}) = \ln (- i \sqrt{2})$

Step 12.3

The equation cannot be solved because $\ln (- i \sqrt{2})$ is undefined.

Undefined

Step 12.4

There is no solution for $e^{x} = - i \sqrt{2}$

No solution

Step 13

List the solutions that makes the equation true.

$x = \ln (2 \sqrt{2}), \ln (i \cdot 2^{\frac{1}{2}})$