Finite Math Examples

$2 e^{2 x} - 5 e^{x} + 4 = 0$

Step 1

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

$2 {(e^{x})}^{2} - 5 e^{x} + 4 = 0$

Step 2

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

$2 u^{2} - 5 u + 4 = 0$

Step 3

Solve for $u$ .

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Step 3.1

Use the quadratic formula to find the solutions.

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

Step 3.2

Substitute the values $a = 2$ , $b = - 5$ , and $c = 4$ into the quadratic formula and solve for $u$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (2 \cdot 4)}}{2 \cdot 2}$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the numerator.

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Step 3.3.1.1

Raise $- 5$ to the power of $2$ .

$u = \frac{5 \pm \sqrt{25 - 4 \cdot 2 \cdot 4}}{2 \cdot 2}$

Step 3.3.1.2

Multiply $- 4 \cdot 2 \cdot 4$ .

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Step 3.3.1.2.1

Multiply $- 4$ by $2$ .

$u = \frac{5 \pm \sqrt{25 - 8 \cdot 4}}{2 \cdot 2}$

Step 3.3.1.2.2

Multiply $- 8$ by $4$ .

$u = \frac{5 \pm \sqrt{25 - 32}}{2 \cdot 2}$

Step 3.3.1.3

Subtract $32$ from $25$ .

$u = \frac{5 \pm \sqrt{- 7}}{2 \cdot 2}$

Step 3.3.1.4

Rewrite $- 7$ as $- 1 (7)$ .

$u = \frac{5 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 2}$

Step 3.3.1.5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$u = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 2}$

Step 3.3.1.6

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

$u = \frac{5 \pm i \sqrt{7}}{2 \cdot 2}$

Step 3.3.2

Multiply $2$ by $2$ .

$u = \frac{5 \pm i \sqrt{7}}{4}$

Step 3.4

The final answer is the combination of both solutions.

$u = \frac{5 + i \sqrt{7}}{4}, \frac{5 - i \sqrt{7}}{4}$

Step 4

Substitute $\frac{5 + i \sqrt{7}}{4}$ for $u$ in $u = e^{x}$ .

$\frac{5 + i \sqrt{7}}{4} = e^{x}$

Step 5

Solve $\frac{5 + i \sqrt{7}}{4} = e^{x}$ .

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Step 5.1

Rewrite the equation as $e^{x} = \frac{5 + i \sqrt{7}}{4}$ .

$e^{x} = \frac{5 + i \sqrt{7}}{4}$

Step 5.2

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

$\ln (e^{x}) = \ln (\frac{5 + i \sqrt{7}}{4})$

Step 5.3

Expand the left side.

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Step 5.3.1

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

$x \ln (e) = \ln (\frac{5 + i \sqrt{7}}{4})$

Step 5.3.2

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

$x \cdot 1 = \ln (\frac{5 + i \sqrt{7}}{4})$

Step 5.3.3

Multiply $x$ by $1$ .

$x = \ln (\frac{5 + i \sqrt{7}}{4})$

Step 5.4

Expand the right side.

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Step 5.4.1

Rewrite $\ln (\frac{5 + i \sqrt{7}}{4})$ as $\ln (5 + i \sqrt{7}) - \ln (4)$ .

$x = \ln (5 + i \sqrt{7}) - \ln (4)$

Step 5.4.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$x = \ln (5 + i \cdot 7^{\frac{1}{2}}) - \ln (4)$

Step 5.4.3

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

$x = \ln (5 + i \cdot 7^{\frac{1}{2}}) - \ln (2^{2})$

Step 5.4.4

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

$x = \ln (5 + i \cdot 7^{\frac{1}{2}}) - (2 \ln (2))$

Step 5.4.5

Multiply $2$ by $- 1$ .

$x = \ln (5 + i \cdot 7^{\frac{1}{2}}) - 2 \ln (2)$

Step 5.5

Simplify.

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Step 5.5.1

Simplify each term.

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Step 5.5.1.1

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

$x = \ln (5 + i \cdot 7^{\frac{1}{2}}) - \ln (2^{2})$

Step 5.5.1.2

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

$x = \ln (5 + i \cdot 7^{\frac{1}{2}}) - \ln (4)$

Step 5.5.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$x = \ln (\frac{5 + i \cdot 7^{\frac{1}{2}}}{4})$

Step 6

Substitute $\frac{5 - i \sqrt{7}}{4}$ for $u$ in $u = e^{x}$ .

$\frac{5 - i \sqrt{7}}{4} = e^{x}$

Step 7

Solve $\frac{5 - i \sqrt{7}}{4} = e^{x}$ .

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Step 7.1

Rewrite the equation as $e^{x} = \frac{5 - i \sqrt{7}}{4}$ .

$e^{x} = \frac{5 - i \sqrt{7}}{4}$

Step 7.2

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

$\ln (e^{x}) = \ln (\frac{5 - i \sqrt{7}}{4})$

Step 7.3

Expand the left side.

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Step 7.3.1

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

$x \ln (e) = \ln (\frac{5 - i \sqrt{7}}{4})$

Step 7.3.2

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

$x \cdot 1 = \ln (\frac{5 - i \sqrt{7}}{4})$

Step 7.3.3

Multiply $x$ by $1$ .

$x = \ln (\frac{5 - i \sqrt{7}}{4})$

Step 7.4

Expand the right side.

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Step 7.4.1

Rewrite $\ln (\frac{5 - i \sqrt{7}}{4})$ as $\ln (5 - i \sqrt{7}) - \ln (4)$ .

$x = \ln (5 - i \sqrt{7}) - \ln (4)$

Step 7.4.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$x = \ln (5 - i \cdot 7^{\frac{1}{2}}) - \ln (4)$

Step 7.4.3

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

$x = \ln (5 - i \cdot 7^{\frac{1}{2}}) - \ln (2^{2})$

Step 7.4.4

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

$x = \ln (5 - i \cdot 7^{\frac{1}{2}}) - (2 \ln (2))$

Step 7.4.5

Multiply $2$ by $- 1$ .

$x = \ln (5 - i \cdot 7^{\frac{1}{2}}) - 2 \ln (2)$

Step 7.5

Simplify.

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Step 7.5.1

Simplify each term.

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Step 7.5.1.1

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

$x = \ln (5 - i \cdot 7^{\frac{1}{2}}) - \ln (2^{2})$

Step 7.5.1.2

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

$x = \ln (5 - i \cdot 7^{\frac{1}{2}}) - \ln (4)$

Step 7.5.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$x = \ln (\frac{5 - i \cdot 7^{\frac{1}{2}}}{4})$

Step 8

List the solutions that makes the equation true.

$x = \ln (\frac{5 + i \cdot 7^{\frac{1}{2}}}{4}), \ln (\frac{5 - i \cdot 7^{\frac{1}{2}}}{4})$