Trigonometry Examples

$e^{2 x} - 7 e^{x} = 8$

Step 1

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

${(e^{x})}^{2} - 7 e^{x} = 8$

Step 2

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

$u^{2} - 7 u = 8$

Step 3

Solve for $u$ .

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Step 3.1

Subtract $8$ from both sides of the equation.

$u^{2} - 7 u - 8 = 0$

Step 3.2

Factor $u^{2} - 7 u - 8$ using the AC method.

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Step 3.2.1

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 $- 8$ and whose sum is $- 7$ .

$- 8, 1$

Step 3.2.2

Write the factored form using these integers.

$(u - 8) (u + 1) = 0$

Step 3.3

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

$u - 8 = 0$

$u + 1 = 0$

Step 3.4

Set $u - 8$ equal to $0$ and solve for $u$ .

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Step 3.4.1

Set $u - 8$ equal to $0$ .

$u - 8 = 0$

Step 3.4.2

Add $8$ to both sides of the equation.

$u = 8$

Step 3.5

Set $u + 1$ equal to $0$ and solve for $u$ .

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Step 3.5.1

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Step 3.5.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 3.6

The final solution is all the values that make $(u - 8) (u + 1) = 0$ true.

$u = 8, - 1$

Step 4

Substitute $8$ for $u$ in $u = e^{x}$ .

$8 = e^{x}$

Step 5

Solve $8 = e^{x}$ .

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Step 5.1

Rewrite the equation as $e^{x} = 8$ .

$e^{x} = 8$

Step 5.2

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

$\ln (e^{x}) = \ln (8)$

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.

$x \ln (e) = \ln (8)$

Step 5.3.2

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

$x \cdot 1 = \ln (8)$

Step 5.3.3

Multiply $x$ by $1$ .

$x = \ln (8)$

Step 6

Substitute $- 1$ for $u$ in $u = e^{x}$ .

$- 1 = e^{x}$

Step 7

Solve $- 1 = e^{x}$ .

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Step 7.1

Rewrite the equation as $e^{x} = - 1$ .

$e^{x} = - 1$

Step 7.2

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

$\ln (e^{x}) = \ln (- 1)$

Step 7.3

The equation cannot be solved because $\ln (- 1)$ is undefined.

Undefined

Step 7.4

There is no solution for $e^{x} = - 1$

No solution

Step 8

List the solutions that makes the equation true.

$x = \ln (8)$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \ln (8)$

Decimal Form:

$x = 2.07944154 \dots$