Algebra Examples

${(\frac{1}{27})}^{x^{2}} = 9^{3 x + 8} \cdot {(\frac{1}{81})}^{x^{2}}$

Step 1

Take the log of both sides of the equation.

$\ln ({(\frac{1}{27})}^{x^{2}}) = \ln (9^{3 x + 8} \cdot {(\frac{1}{81})}^{x^{2}})$

Step 2

Expand $\ln ({(\frac{1}{27})}^{x^{2}})$ by moving $x^{2}$ outside the logarithm.

$x^{2} \ln (\frac{1}{27}) = \ln (9^{3 x + 8} \cdot {(\frac{1}{81})}^{x^{2}})$

Step 3

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

$x^{2} (\ln (1) - \ln (27)) = \ln (9^{3 x + 8} \cdot {(\frac{1}{81})}^{x^{2}})$

Step 4

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

$x^{2} (0 - \ln (27)) = \ln (9^{3 x + 8} \cdot {(\frac{1}{81})}^{x^{2}})$

Step 5

Rewrite $\ln (27)$ as $\ln (3^{3})$ .

$x^{2} (0 - \ln (3^{3})) = \ln (9^{3 x + 8} \cdot {(\frac{1}{81})}^{x^{2}})$

Step 6

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

$x^{2} (0 - (3 \ln (3))) = \ln (9^{3 x + 8} \cdot {(\frac{1}{81})}^{x^{2}})$

Step 7

Multiply $3$ by $- 1$ .

$x^{2} (0 - 3 \ln (3)) = \ln (9^{3 x + 8} \cdot {(\frac{1}{81})}^{x^{2}})$

Step 8

Subtract $3 \ln (3)$ from $0$ .

$x^{2} (- 3 \ln (3)) = \ln (9^{3 x + 8} \cdot {(\frac{1}{81})}^{x^{2}})$

Step 9

Rewrite $\ln (9^{3 x + 8} \cdot {(\frac{1}{81})}^{x^{2}})$ as $\ln (9^{3 x + 8}) + \ln ({(\frac{1}{81})}^{x^{2}})$ .

$x^{2} (- 3 \ln (3)) = \ln (9^{3 x + 8}) + \ln ({(\frac{1}{81})}^{x^{2}})$

Step 10

Expand $\ln (9^{3 x + 8})$ by moving $3 x + 8$ outside the logarithm.

$x^{2} (- 3 \ln (3)) = (3 x + 8) \ln (9) + \ln ({(\frac{1}{81})}^{x^{2}})$

Step 11

Expand $\ln ({(\frac{1}{81})}^{x^{2}})$ by moving $x^{2}$ outside the logarithm.

$x^{2} (- 3 \ln (3)) = (3 x + 8) \ln (9) + x^{2} \ln (\frac{1}{81})$

Step 12

Rewrite $\ln (\frac{1}{81})$ as $\ln (1) - \ln (81)$ .

$x^{2} (- 3 \ln (3)) = (3 x + 8) \ln (9) + x^{2} (\ln (1) - \ln (81))$

Step 13

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

$x^{2} (- 3 \ln (3)) = (3 x + 8) \ln (9) + x^{2} (0 - \ln (81))$

Step 14

Rewrite $\ln (81)$ as $\ln (3^{4})$ .

$x^{2} (- 3 \ln (3)) = (3 x + 8) \ln (9) + x^{2} (0 - \ln (3^{4}))$

Step 15

Expand $\ln (3^{4})$ by moving $4$ outside the logarithm.

$x^{2} (- 3 \ln (3)) = (3 x + 8) \ln (9) + x^{2} (0 - (4 \ln (3)))$

Step 16

Multiply $4$ by $- 1$ .

$x^{2} (- 3 \ln (3)) = (3 x + 8) \ln (9) + x^{2} (0 - 4 \ln (3))$

Step 17

Subtract $4 \ln (3)$ from $0$ .

$x^{2} (- 3 \ln (3)) = (3 x + 8) \ln (9) + x^{2} (- 4 \ln (3))$

Step 18

Solve the equation for $x$ .

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Step 18.1

Simplify the left side.

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Step 18.1.1

Simplify $x^{2} (- 3 \ln (3))$ .

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Step 18.1.1.1

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

$x^{2} (- \ln (3^{3})) = (3 x + 8) \ln (9) + x^{2} (- 4 \ln (3))$

Step 18.1.1.2

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

$x^{2} (- \ln (27)) = (3 x + 8) \ln (9) + x^{2} (- 4 \ln (3))$

Step 18.1.1.3

Reorder factors in $x^{2} (- \ln (27))$ .

$- x^{2} \ln (27) = (3 x + 8) \ln (9) + x^{2} (- 4 \ln (3))$

Step 18.2

Simplify the right side.

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Step 18.2.1

Simplify $(3 x + 8) \ln (9) + x^{2} (- 4 \ln (3))$ .

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Step 18.2.1.1

Simplify each term.

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Step 18.2.1.1.1

Apply the distributive property.

$- x^{2} \ln (27) = 3 x \ln (9) + 8 \ln (9) + x^{2} (- 4 \ln (3))$

Step 18.2.1.1.2

Simplify $3 \ln (9)$ by moving $3$ inside the logarithm.

$- x^{2} \ln (27) = x \ln (9^{3}) + 8 \ln (9) + x^{2} (- 4 \ln (3))$

Step 18.2.1.1.3

Simplify $8 \ln (9)$ by moving $8$ inside the logarithm.

$- x^{2} \ln (27) = x \ln (9^{3}) + \ln (9^{8}) + x^{2} (- 4 \ln (3))$

Step 18.2.1.1.4

Simplify each term.

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Step 18.2.1.1.4.1

Raise $9$ to the power of $3$ .

$- x^{2} \ln (27) = x \ln (729) + \ln (9^{8}) + x^{2} (- 4 \ln (3))$

Step 18.2.1.1.4.2

Raise $9$ to the power of $8$ .

$- x^{2} \ln (27) = x \ln (729) + \ln (43046721) + x^{2} (- 4 \ln (3))$

Step 18.2.1.1.5

Simplify $- 4 \ln (3)$ by moving $4$ inside the logarithm.

$- x^{2} \ln (27) = x \ln (729) + \ln (43046721) + x^{2} (- \ln (3^{4}))$

Step 18.2.1.1.6

Raise $3$ to the power of $4$ .

$- x^{2} \ln (27) = x \ln (729) + \ln (43046721) + x^{2} (- \ln (81))$

Step 18.2.1.2

Reorder factors in $x \ln (729) + \ln (43046721) + x^{2} (- \ln (81))$ .

$- x^{2} \ln (27) = x \ln (729) + \ln (43046721) - x^{2} \ln (81)$

Step 18.3

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

$- x^{2} \ln (27) - x \ln (729) - \ln (43046721) + x^{2} \ln (81) = 0$

Step 18.4

Use the quadratic formula to find the solutions.

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

Step 18.5

Substitute the values $a = - \ln (27) + \ln (81)$ , $b = - \ln (729)$ , and $c = - \ln (43046721)$ into the quadratic formula and solve for $x$ .

$\frac{\ln (729) \pm \sqrt{{(- \ln (729))}^{2} - 4 \cdot ((- \ln (27) + \ln (81)) \cdot (- \ln (43046721)))}}{2 (- \ln (27) + \ln (81))}$

Step 18.6

Simplify the numerator.

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Step 18.6.1

Apply the product rule to $- \ln (729)$ .

$x = \frac{\ln (729) \pm \sqrt{{(- 1)}^{2} \ln^{2} (729) - 4 \cdot (- \ln (27) + \ln (81)) \cdot (- 1 \ln (43046721))}}{2 (- \ln (27) + \ln (81))}$

Step 18.6.2

Raise $- 1$ to the power of $2$ .

$x = \frac{\ln (729) \pm \sqrt{1 \ln^{2} (729) - 4 \cdot (- \ln (27) + \ln (81)) \cdot (- 1 \ln (43046721))}}{2 (- \ln (27) + \ln (81))}$

Step 18.6.3

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

$x = \frac{\ln (729) \pm \sqrt{\ln^{2} (729) - 4 \cdot (- \ln (27) + \ln (81)) \cdot (- 1 \ln (43046721))}}{2 (- \ln (27) + \ln (81))}$

Step 18.6.4

Apply the distributive property.

$x = \frac{\ln (729) \pm \sqrt{\ln^{2} (729) + (- 4 (- \ln (27)) - 4 \ln (81)) \cdot (- 1 \ln (43046721))}}{2 (- \ln (27) + \ln (81))}$

Step 18.6.5

Multiply $- 1$ by $- 4$ .

$x = \frac{\ln (729) \pm \sqrt{\ln^{2} (729) + (4 \ln (27) - 4 \ln (81)) \cdot (- 1 \ln (43046721))}}{2 (- \ln (27) + \ln (81))}$

Step 18.6.6

Apply the distributive property.

$x = \frac{\ln (729) \pm \sqrt{\ln^{2} (729) + (4 \ln (27) \cdot - 1 - 4 \ln (81) \cdot - 1) \ln (43046721)}}{2 (- \ln (27) + \ln (81))}$

Step 18.6.7

Multiply $- 1$ by $4$ .

$x = \frac{\ln (729) \pm \sqrt{\ln^{2} (729) + (- 4 \ln (27) - 4 \ln (81) \cdot - 1) \ln (43046721)}}{2 (- \ln (27) + \ln (81))}$

Step 18.6.8

Multiply $- 1$ by $- 4$ .

$x = \frac{\ln (729) \pm \sqrt{\ln^{2} (729) + (- 4 \ln (27) + 4 \ln (81)) \ln (43046721)}}{2 (- \ln (27) + \ln (81))}$

Step 18.6.9

Apply the distributive property.

$x = \frac{\ln (729) \pm \sqrt{\ln^{2} (729) - 4 \ln (27) \ln (43046721) + 4 \ln (81) \ln (43046721)}}{2 (- \ln (27) + \ln (81))}$

Step 18.7

The final answer is the combination of both solutions.

$x = - \frac{\ln (729) + \sqrt{\ln^{2} (729) - 4 \ln (27) \ln (43046721) + 4 \ln (81) \ln (43046721)}}{2 (\ln (27) - \ln (81))}, - \frac{\ln (729) - \sqrt{\ln^{2} (729) - 4 \ln (27) \ln (43046721) + 4 \ln (81) \ln (43046721)}}{2 (\ln (27) - \ln (81))}$

Step 19

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

$x = 8, - 2$