Precalculus Examples

$\frac{3^{x} - 3^{- x}}{4} = 5$

Step 1

Multiply both sides by $4$ .

$\frac{3^{x} - 3^{- x}}{4} \cdot 4 = 5 \cdot 4$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $4$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{3^{x} - 3^{- x}}{4} \cdot 4 = 5 \cdot 4$

Step 2.1.1.2

Rewrite the expression.

$3^{x} - 3^{- x} = 5 \cdot 4$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $5$ by $4$ .

$3^{x} - 3^{- x} = 20$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite $3^{- x}$ as exponentiation.

$3^{x} - {(3^{x})}^{- 1} = 20$

Step 3.2

Substitute $u$ for $3^{x}$ .

$u - u^{- 1} = 20$

Step 3.3

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

$u - \frac{1}{u} = 20$

Step 3.4

Solve for $u$ .

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Step 3.4.1

Find the LCD of the terms in the equation.

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Step 3.4.1.1

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 3.4.1.2

The LCM of one and any expression is the expression.

$u$

Step 3.4.2

Multiply each term in $u - \frac{1}{u} = 20$ by $u$ to eliminate the fractions.

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Step 3.4.2.1

Multiply each term in $u - \frac{1}{u} = 20$ by $u$ .

$u \cdot u - \frac{1}{u} u = 20 u$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Simplify each term.

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Step 3.4.2.2.1.1

Multiply $u$ by $u$ .

$u^{2} - \frac{1}{u} u = 20 u$

Step 3.4.2.2.1.2

Cancel the common factor of $u$ .

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Step 3.4.2.2.1.2.1

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

$u^{2} + \frac{- 1}{u} u = 20 u$

Step 3.4.2.2.1.2.2

Cancel the common factor.

$u^{2} + \frac{- 1}{u} u = 20 u$

Step 3.4.2.2.1.2.3

Rewrite the expression.

$u^{2} - 1 = 20 u$

Step 3.4.3

Solve the equation.

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Step 3.4.3.1

Subtract $20 u$ from both sides of the equation.

$u^{2} - 1 - 20 u = 0$

Step 3.4.3.2

Use the quadratic formula to find the solutions.

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

Step 3.4.3.3

Substitute the values $a = 1$ , $b = - 20$ , and $c = - 1$ into the quadratic formula and solve for $u$ .

$\frac{20 \pm \sqrt{{(- 20)}^{2} - 4 \cdot (1 \cdot - 1)}}{2 \cdot 1}$

Step 3.4.3.4

Simplify.

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Step 3.4.3.4.1

Simplify the numerator.

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Step 3.4.3.4.1.1

Raise $- 20$ to the power of $2$ .

$u = \frac{20 \pm \sqrt{400 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 3.4.3.4.1.2

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

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Step 3.4.3.4.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{20 \pm \sqrt{400 - 4 \cdot - 1}}{2 \cdot 1}$

Step 3.4.3.4.1.2.2

Multiply $- 4$ by $- 1$ .

$u = \frac{20 \pm \sqrt{400 + 4}}{2 \cdot 1}$

Step 3.4.3.4.1.3

Add $400$ and $4$ .

$u = \frac{20 \pm \sqrt{404}}{2 \cdot 1}$

Step 3.4.3.4.1.4

Rewrite $404$ as $2^{2} \cdot 101$ .

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Step 3.4.3.4.1.4.1

Factor $4$ out of $404$ .

$u = \frac{20 \pm \sqrt{4 (101)}}{2 \cdot 1}$

Step 3.4.3.4.1.4.2

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

$u = \frac{20 \pm \sqrt{2^{2} \cdot 101}}{2 \cdot 1}$

Step 3.4.3.4.1.5

Pull terms out from under the radical.

$u = \frac{20 \pm 2 \sqrt{101}}{2 \cdot 1}$

Step 3.4.3.4.2

Multiply $2$ by $1$ .

$u = \frac{20 \pm 2 \sqrt{101}}{2}$

Step 3.4.3.4.3

Simplify $\frac{20 \pm 2 \sqrt{101}}{2}$ .

$u = 10 \pm \sqrt{101}$

Step 3.4.3.5

The final answer is the combination of both solutions.

$u = 10 + \sqrt{101}, 10 - \sqrt{101}$

Step 3.5

Substitute $10 + \sqrt{101}$ for $u$ in $u = 3^{x}$ .

$10 + \sqrt{101} = 3^{x}$

Step 3.6

Solve $10 + \sqrt{101} = 3^{x}$ .

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Step 3.6.1

Rewrite the equation as $3^{x} = 10 + \sqrt{101}$ .

$3^{x} = 10 + \sqrt{101}$

Step 3.6.2

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

$\ln (3^{x}) = \ln (10 + \sqrt{101})$

Step 3.6.3

Expand $\ln (3^{x})$ by moving $x$ outside the logarithm.

$x \ln (3) = \ln (10 + \sqrt{101})$

Step 3.6.4

Divide each term in $x \ln (3) = \ln (10 + \sqrt{101})$ by $\ln (3)$ and simplify.

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Step 3.6.4.1

Divide each term in $x \ln (3) = \ln (10 + \sqrt{101})$ by $\ln (3)$ .

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (10 + \sqrt{101})}{\ln (3)}$

Step 3.6.4.2

Simplify the left side.

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Step 3.6.4.2.1

Cancel the common factor of $\ln (3)$ .

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Step 3.6.4.2.1.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (10 + \sqrt{101})}{\ln (3)}$

Step 3.6.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (10 + \sqrt{101})}{\ln (3)}$

Step 3.7

Substitute $10 - \sqrt{101}$ for $u$ in $u = 3^{x}$ .

$10 - \sqrt{101} = 3^{x}$

Step 3.8

Solve $10 - \sqrt{101} = 3^{x}$ .

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Step 3.8.1

Rewrite the equation as $3^{x} = 10 - \sqrt{101}$ .

$3^{x} = 10 - \sqrt{101}$

Step 3.8.2

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

$\ln (3^{x}) = \ln (10 - \sqrt{101})$

Step 3.8.3

The equation cannot be solved because $\ln (10 - \sqrt{101})$ is undefined.

Undefined

Step 3.8.4

There is no solution for $3^{x} = 10 - \sqrt{101}$

No solution

Step 3.9

List the solutions that makes the equation true.

$x = \frac{\ln (10 + \sqrt{101})}{\ln (3)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (10 + \sqrt{101})}{\ln (3)}$

Decimal Form:

$x = 2.72910013 \dots$