Precalculus Examples

$4^{x} - 4^{- x} = 2$

Step 1

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

$4^{x} - {(4^{x})}^{- 1} = 2$

Step 2

Substitute $u$ for $4^{x}$ .

$u - u^{- 1} = 2$

Step 3

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

$u - \frac{1}{u} = 2$

Step 4

Solve for $u$ .

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Step 4.1

Find the LCD of the terms in the equation.

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Step 4.1.1

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

$1, u, 1$

Step 4.1.2

The LCM of one and any expression is the expression.

$u$

Step 4.2

Multiply each term in $u - \frac{1}{u} = 2$ by $u$ to eliminate the fractions.

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Step 4.2.1

Multiply each term in $u - \frac{1}{u} = 2$ by $u$ .

$u \cdot u - \frac{1}{u} u = 2 u$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Multiply $u$ by $u$ .

$u^{2} - \frac{1}{u} u = 2 u$

Step 4.2.2.1.2

Cancel the common factor of $u$ .

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Step 4.2.2.1.2.1

Move the leading negative in $- \frac{1}{u}$ into the numerator.

$u^{2} + \frac{- 1}{u} u = 2 u$

Step 4.2.2.1.2.2

Cancel the common factor.

$u^{2} + \frac{- 1}{u} u = 2 u$

Step 4.2.2.1.2.3

Rewrite the expression.

$u^{2} - 1 = 2 u$

Step 4.3

Solve the equation.

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Step 4.3.1

Subtract $2 u$ from both sides of the equation.

$u^{2} - 1 - 2 u = 0$

Step 4.3.2

Use the quadratic formula to find the solutions.

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

Step 4.3.3

Substitute the values $a = 1$ , $b = - 2$ , and $c = - 1$ into the quadratic formula and solve for $u$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot - 1)}}{2 \cdot 1}$

Step 4.3.4

Simplify.

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Step 4.3.4.1

Simplify the numerator.

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Step 4.3.4.1.1

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

$u = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 4.3.4.1.2

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 4.3.4.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{2 \pm \sqrt{4 - 4 \cdot - 1}}{2 \cdot 1}$

Step 4.3.4.1.2.2

Multiply $- 4$ by $- 1$ .

$u = \frac{2 \pm \sqrt{4 + 4}}{2 \cdot 1}$

Step 4.3.4.1.3

Add $4$ and $4$ .

$u = \frac{2 \pm \sqrt{8}}{2 \cdot 1}$

Step 4.3.4.1.4

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

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Step 4.3.4.1.4.1

Factor $4$ out of $8$ .

$u = \frac{2 \pm \sqrt{4 (2)}}{2 \cdot 1}$

Step 4.3.4.1.4.2

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

$u = \frac{2 \pm \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 4.3.4.1.5

Pull terms out from under the radical.

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

Step 4.3.4.2

Multiply $2$ by $1$ .

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

Step 4.3.4.3

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

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

Step 4.3.5

The final answer is the combination of both solutions.

$u = 1 + \sqrt{2}, 1 - \sqrt{2}$

Step 5

Substitute $1 + \sqrt{2}$ for $u$ in $u = 4^{x}$ .

$1 + \sqrt{2} = 4^{x}$

Step 6

Solve $1 + \sqrt{2} = 4^{x}$ .

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Step 6.1

Rewrite the equation as $4^{x} = 1 + \sqrt{2}$ .

$4^{x} = 1 + \sqrt{2}$

Step 6.2

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

$\ln (4^{x}) = \ln (1 + \sqrt{2})$

Step 6.3

Expand $\ln (4^{x})$ by moving $x$ outside the logarithm.

$x \ln (4) = \ln (1 + \sqrt{2})$

Step 6.4

Divide each term in $x \ln (4) = \ln (1 + \sqrt{2})$ by $\ln (4)$ and simplify.

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Step 6.4.1

Divide each term in $x \ln (4) = \ln (1 + \sqrt{2})$ by $\ln (4)$ .

$\frac{x \ln (4)}{\ln (4)} = \frac{\ln (1 + \sqrt{2})}{\ln (4)}$

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Cancel the common factor of $\ln (4)$ .

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Step 6.4.2.1.1

Cancel the common factor.

$\frac{x \ln (4)}{\ln (4)} = \frac{\ln (1 + \sqrt{2})}{\ln (4)}$

Step 6.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (1 + \sqrt{2})}{\ln (4)}$

Step 7

Substitute $1 - \sqrt{2}$ for $u$ in $u = 4^{x}$ .

$1 - \sqrt{2} = 4^{x}$

Step 8

Solve $1 - \sqrt{2} = 4^{x}$ .

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Step 8.1

Rewrite the equation as $4^{x} = 1 - \sqrt{2}$ .

$4^{x} = 1 - \sqrt{2}$

Step 8.2

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

$\ln (4^{x}) = \ln (1 - \sqrt{2})$

Step 8.3

The equation cannot be solved because $\ln (1 - \sqrt{2})$ is undefined.

Undefined

Step 8.4

There is no solution for $4^{x} = 1 - \sqrt{2}$

No solution

Step 9

List the solutions that makes the equation true.

$x = \frac{\ln (1 + \sqrt{2})}{\ln (4)}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (1 + \sqrt{2})}{\ln (4)}$

Decimal Form:

$x = 0.63577665 \dots$