Finite Math Examples

$\frac{1}{27^{2 x}} \cdot 3^{- 2 x} = 81^{x^{2} - 2}$

Step 1

Take the log of both sides of the equation.

$\ln (\frac{1}{27^{2 x}} \cdot 3^{- 2 x}) = \ln (81^{x^{2} - 2})$

Step 2

Rewrite $\ln (\frac{1}{27^{2 x}} \cdot 3^{- 2 x})$ as $\ln (\frac{1}{27^{2 x}}) + \ln (3^{- 2 x})$ .

$\ln (\frac{1}{27^{2 x}}) + \ln (3^{- 2 x}) = \ln (81^{x^{2} - 2})$

Step 3

Rewrite $\ln (\frac{1}{27^{2 x}})$ as $\ln (1) - \ln (27^{2 x})$ .

$\ln (1) - \ln (27^{2 x}) + \ln (3^{- 2 x}) = \ln (81^{x^{2} - 2})$

Step 4

Expand $\ln (27^{2 x})$ by moving $2 x$ outside the logarithm.

$\ln (1) - (2 x \ln (27)) + \ln (3^{- 2 x}) = \ln (81^{x^{2} - 2})$

Step 5

Expand $\ln (3^{- 2 x})$ by moving $- 2 x$ outside the logarithm.

$\ln (1) - (2 x \ln (27)) - 2 x \ln (3) = \ln (81^{x^{2} - 2})$

Step 6

The natural logarithm of $1$ is $0$ .

$0 - (2 x \ln (27)) - 2 x \ln (3) = \ln (81^{x^{2} - 2})$

Step 7

Subtract $2 x \ln (27)$ from $0$ .

$- (2 x \ln (27)) - 2 x \ln (3) = \ln (81^{x^{2} - 2})$

Step 8

Multiply $2$ by $- 1$ .

$- 2 (x \ln (27)) - 2 x \ln (3) = \ln (81^{x^{2} - 2})$

Step 9

Remove parentheses.

$- 2 x \ln (27) - 2 x \ln (3) = \ln (81^{x^{2} - 2})$

Step 10

Expand $\ln (81^{x^{2} - 2})$ by moving $x^{2} - 2$ outside the logarithm.

$- 2 x \ln (27) - 2 x \ln (3) = (x^{2} - 2) \ln (81)$

Step 11

Solve the equation for $x$ .

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Step 11.1

Apply the distributive property.

$- 2 x \ln (27) - 2 x \ln (3) = x^{2} \ln (81) - 2 \ln (81)$

Step 11.2

Simplify the left side.

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Step 11.2.1

Simplify $- 2 x \ln (27) - 2 x \ln (3)$ .

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Step 11.2.1.1

Simplify each term.

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Step 11.2.1.1.1

Simplify $- 2 \ln (27)$ by moving $2$ inside the logarithm.

$x (- \ln (27^{2})) - 2 x \ln (3) = x^{2} \ln (81) - 2 \ln (81)$

Step 11.2.1.1.2

Raise $27$ to the power of $2$ .

$x (- \ln (729)) - 2 x \ln (3) = x^{2} \ln (81) - 2 \ln (81)$

Step 11.2.1.1.3

Simplify $- 2 \ln (3)$ by moving $2$ inside the logarithm.

$x (- \ln (729)) + x (- \ln (3^{2})) = x^{2} \ln (81) - 2 \ln (81)$

Step 11.2.1.1.4

Raise $3$ to the power of $2$ .

$x (- \ln (729)) + x (- \ln (9)) = x^{2} \ln (81) - 2 \ln (81)$

Step 11.2.1.2

Reorder factors in $x (- \ln (729)) + x (- \ln (9))$ .

$- x \ln (729) - x \ln (9) = x^{2} \ln (81) - 2 \ln (81)$

Step 11.3

Simplify the right side.

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Step 11.3.1

Simplify each term.

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Step 11.3.1.1

Simplify $- 2 \ln (81)$ by moving $2$ inside the logarithm.

$- x \ln (729) - x \ln (9) = x^{2} \ln (81) - \ln (81^{2})$

Step 11.3.1.2

Raise $81$ to the power of $2$ .

$- x \ln (729) - x \ln (9) = x^{2} \ln (81) - \ln (6561)$

Step 11.4

Move all the terms containing a logarithm to the left side of the equation.

$- x \ln (729) - x \ln (9) - x^{2} \ln (81) + \ln (6561) = 0$

Step 11.5

Use the quadratic formula to find the solutions.

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

Step 11.6

Substitute the values $a = - \ln (81)$ , $b = - \ln (729) - \ln (9)$ , and $c = \ln (6561)$ into the quadratic formula and solve for $x$ .

$\frac{- (- \ln (729) - \ln (9)) \pm \sqrt{{(- \ln (729) - \ln (9))}^{2} - 4 \cdot (- \ln (81) \cdot \ln (6561))}}{2 (- \ln (81))}$

Step 11.7

Simplify.

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Step 11.7.1

Simplify the numerator.

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Step 11.7.1.1

Apply the distributive property.

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{{(- \ln (729) - \ln (9))}^{2} - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.2

Multiply $- - \ln (729)$ .

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Step 11.7.1.2.1

Multiply $- 1$ by $- 1$ .

$x = \frac{1 \ln (729) + \ln (9) \pm \sqrt{{(- \ln (729) - \ln (9))}^{2} - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.2.2

Multiply $\ln (729)$ by $1$ .

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{{(- \ln (729) - \ln (9))}^{2} - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.3

Multiply $- - \ln (9)$ .

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Step 11.7.1.3.1

Multiply $- 1$ by $- 1$ .

$x = \frac{\ln (729) + 1 \ln (9) \pm \sqrt{{(- \ln (729) - \ln (9))}^{2} - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.3.2

Multiply $\ln (9)$ by $1$ .

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{{(- \ln (729) - \ln (9))}^{2} - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.4

Rewrite ${(- \ln (729) - \ln (9))}^{2}$ as $(- \ln (729) - \ln (9)) (- \ln (729) - \ln (9))$ .

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{(- \ln (729) - \ln (9)) (- \ln (729) - \ln (9)) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.5

Expand $(- \ln (729) - \ln (9)) (- \ln (729) - \ln (9))$ using the FOIL Method.

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Step 11.7.1.5.1

Apply the distributive property.

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{- \ln (729) (- \ln (729) - \ln (9)) - \ln (9) (- \ln (729) - \ln (9)) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.5.2

Apply the distributive property.

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{- \ln (729) (- \ln (729)) - \ln (729) (- \ln (9)) - \ln (9) (- \ln (729) - \ln (9)) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.5.3

Apply the distributive property.

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{- \ln (729) (- \ln (729)) - \ln (729) (- \ln (9)) - \ln (9) (- \ln (729)) - \ln (9) (- \ln (9)) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.6

Simplify and combine like terms.

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Step 11.7.1.6.1

Simplify each term.

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Step 11.7.1.6.1.1

Rewrite using the commutative property of multiplication.

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{- 1 \cdot (- 1 \ln (729) \ln (729)) - \ln (729) (- \ln (9)) - \ln (9) (- \ln (729)) - \ln (9) (- \ln (9)) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.6.1.2

Multiply $\ln (729)$ by $\ln (729)$ by adding the exponents.

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Step 11.7.1.6.1.2.1

Move $\ln (729)$ .

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{- 1 \cdot (- 1 (\ln (729) \ln (729))) - \ln (729) (- \ln (9)) - \ln (9) (- \ln (729)) - \ln (9) (- \ln (9)) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.6.1.2.2

Multiply $\ln (729)$ by $\ln (729)$ .

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{- 1 \cdot (- 1 \ln^{2} (729)) - \ln (729) (- \ln (9)) - \ln (9) (- \ln (729)) - \ln (9) (- \ln (9)) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.6.1.3

Multiply $- 1$ by $- 1$ .

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{1 \ln^{2} (729) - \ln (729) (- \ln (9)) - \ln (9) (- \ln (729)) - \ln (9) (- \ln (9)) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.6.1.4

Multiply $\ln^{2} (729)$ by $1$ .

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{\ln^{2} (729) - \ln (729) (- \ln (9)) - \ln (9) (- \ln (729)) - \ln (9) (- \ln (9)) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.6.1.5

Rewrite using the commutative property of multiplication.

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{\ln^{2} (729) - 1 \cdot (- 1 \ln (729) \ln (9)) - \ln (9) (- \ln (729)) - \ln (9) (- \ln (9)) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.6.1.6

Multiply $- 1$ by $- 1$ .

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{\ln^{2} (729) + 1 \ln (729) \ln (9) - \ln (9) (- \ln (729)) - \ln (9) (- \ln (9)) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.6.1.7

Multiply $\ln (729)$ by $1$ .

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{\ln^{2} (729) + \ln (729) \ln (9) - \ln (9) (- \ln (729)) - \ln (9) (- \ln (9)) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.6.1.8

Rewrite using the commutative property of multiplication.

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{\ln^{2} (729) + \ln (729) \ln (9) - 1 \cdot (- 1 \ln (9) \ln (729)) - \ln (9) (- \ln (9)) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.6.1.9

Multiply $- 1$ by $- 1$ .

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{\ln^{2} (729) + \ln (729) \ln (9) + 1 \ln (9) \ln (729) - \ln (9) (- \ln (9)) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.6.1.10

Multiply $\ln (9)$ by $1$ .

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{\ln^{2} (729) + \ln (729) \ln (9) + \ln (9) \ln (729) - \ln (9) (- \ln (9)) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.6.1.11

Rewrite using the commutative property of multiplication.

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{\ln^{2} (729) + \ln (729) \ln (9) + \ln (9) \ln (729) - 1 \cdot (- 1 \ln (9) \ln (9)) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.6.1.12

Multiply $\ln (9)$ by $\ln (9)$ by adding the exponents.

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Step 11.7.1.6.1.12.1

Move $\ln (9)$ .

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{\ln^{2} (729) + \ln (729) \ln (9) + \ln (9) \ln (729) - 1 \cdot (- 1 (\ln (9) \ln (9))) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.6.1.12.2

Multiply $\ln (9)$ by $\ln (9)$ .

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{\ln^{2} (729) + \ln (729) \ln (9) + \ln (9) \ln (729) - 1 \cdot (- 1 \ln^{2} (9)) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.6.1.13

Multiply $- 1$ by $- 1$ .

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{\ln^{2} (729) + \ln (729) \ln (9) + \ln (9) \ln (729) + 1 \ln^{2} (9) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.6.1.14

Multiply $\ln^{2} (9)$ by $1$ .

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{\ln^{2} (729) + \ln (729) \ln (9) + \ln (9) \ln (729) + \ln^{2} (9) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.6.2

Add $\ln (729) \ln (9)$ and $\ln (9) \ln (729)$ .

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Step 11.7.1.6.2.1

Reorder $\ln (729)$ and $\ln (9)$ .

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{\ln^{2} (729) + \ln (9) \ln (729) + \ln (9) \ln (729) + \ln^{2} (9) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.6.2.2

Add $\ln (9) \ln (729)$ and $\ln (9) \ln (729)$ .

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{\ln^{2} (729) + 2 \ln (9) \ln (729) + \ln^{2} (9) - 4 \cdot (- 1 \ln (81)) \cdot \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.1.7

Multiply $- 4$ by $- 1$ .

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{\ln^{2} (729) + 2 \ln (9) \ln (729) + \ln^{2} (9) + 4 \ln (81) \ln (6561)}}{2 (- \ln (81))}$

Step 11.7.2

Multiply $- 1$ by $2$ .

$x = \frac{\ln (729) + \ln (9) \pm \sqrt{\ln^{2} (729) + 2 \ln (9) \ln (729) + \ln^{2} (9) + 4 \ln (81) \ln (6561)}}{- 2 \ln (81)}$

Step 11.7.3

Move the negative in front of the fraction.

$x = - \frac{\ln (729) + \ln (9) \pm \sqrt{\ln^{2} (729) + 2 \ln (9) \ln (729) + \ln^{2} (9) + 4 \ln (81) \ln (6561)}}{2 \ln (81)}$

Step 11.8

The final answer is the combination of both solutions.

$x = - \frac{\ln (729) + \ln (9) + \sqrt{\ln^{2} (729) + 2 \ln (9) \ln (729) + \ln^{2} (9) + 4 \ln (81) \ln (6561)}}{2 \ln (81)}, - \frac{\ln (729) + \ln (9) - \sqrt{\ln^{2} (729) + 2 \ln (9) \ln (729) + \ln^{2} (9) + 4 \ln (81) \ln (6561)}}{2 \ln (81)}$

Step 12

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

$x = - 2.73205080 \dots, 0.73205080 \dots$