Precalculus Examples

$g (x) = e^{x} + 12 e^{- x} - 7$

Step 1

Set $e^{x} + 12 e^{- x} - 7$ equal to $0$ .

$e^{x} + 12 e^{- x} - 7 = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$e^{x} + 12 {(e^{x})}^{- 1} - 7 = 0$

Step 2.2

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

$u + 12 u^{- 1} - 7 = 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$u + 12 (\frac{1}{u}) - 7 = 0$

Step 2.3.2

Combine $12$ and $\frac{1}{u}$ .

$u + \frac{12}{u} - 7 = 0$

Step 2.4

Solve for $u$ .

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Step 2.4.1

Find the LCD of the terms in the equation.

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Step 2.4.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, u, 1, 1$

Step 2.4.1.2

The LCM of one and any expression is the expression.

$u$

Step 2.4.2

Multiply each term in $u + \frac{12}{u} - 7 = 0$ by $u$ to eliminate the fractions.

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Step 2.4.2.1

Multiply each term in $u + \frac{12}{u} - 7 = 0$ by $u$ .

$u \cdot u + \frac{12}{u} u - 7 u = 0 u$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Simplify each term.

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Step 2.4.2.2.1.1

Multiply $u$ by $u$ .

$u^{2} + \frac{12}{u} u - 7 u = 0 u$

Step 2.4.2.2.1.2

Cancel the common factor of $u$ .

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Step 2.4.2.2.1.2.1

Cancel the common factor.

$u^{2} + \frac{12}{u} u - 7 u = 0 u$

Step 2.4.2.2.1.2.2

Rewrite the expression.

$u^{2} + 12 - 7 u = 0 u$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Multiply $0$ by $u$ .

$u^{2} + 12 - 7 u = 0$

Step 2.4.3

Solve the equation.

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Step 2.4.3.1

Factor $u^{2} + 12 - 7 u$ using the AC method.

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Step 2.4.3.1.1

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 $12$ and whose sum is $- 7$ .

$- 4, - 3$

Step 2.4.3.1.2

Write the factored form using these integers.

$(u - 4) (u - 3) = 0$

Step 2.4.3.2

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

$u - 4 = 0$

$u - 3 = 0$

Step 2.4.3.3

Set $u - 4$ equal to $0$ and solve for $u$ .

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Step 2.4.3.3.1

Set $u - 4$ equal to $0$ .

$u - 4 = 0$

Step 2.4.3.3.2

Add $4$ to both sides of the equation.

$u = 4$

Step 2.4.3.4

Set $u - 3$ equal to $0$ and solve for $u$ .

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Step 2.4.3.4.1

Set $u - 3$ equal to $0$ .

$u - 3 = 0$

Step 2.4.3.4.2

Add $3$ to both sides of the equation.

$u = 3$

Step 2.4.3.5

The final solution is all the values that make $(u - 4) (u - 3) = 0$ true.

$u = 4, 3$

Step 2.5

Substitute $4$ for $u$ in $u = e^{x}$ .

$4 = e^{x}$

Step 2.6

Solve $4 = e^{x}$ .

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Step 2.6.1

Rewrite the equation as $e^{x} = 4$ .

$e^{x} = 4$

Step 2.6.2

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

$\ln (e^{x}) = \ln (4)$

Step 2.6.3

Expand the left side.

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Step 2.6.3.1

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

$x \ln (e) = \ln (4)$

Step 2.6.3.2

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

$x \cdot 1 = \ln (4)$

Step 2.6.3.3

Multiply $x$ by $1$ .

$x = \ln (4)$

Step 2.7

Substitute $3$ for $u$ in $u = e^{x}$ .

$3 = e^{x}$

Step 2.8

Solve $3 = e^{x}$ .

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Step 2.8.1

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

$e^{x} = 3$

Step 2.8.2

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

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

Step 2.8.3

Expand the left side.

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Step 2.8.3.1

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

$x \ln (e) = \ln (3)$

Step 2.8.3.2

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

$x \cdot 1 = \ln (3)$

Step 2.8.3.3

Multiply $x$ by $1$ .

$x = \ln (3)$

Step 2.9

List the solutions that makes the equation true.

$x = \ln (4), \ln (3)$

Step 3