Precalculus Examples

$7^{x^{2}} = 8^{9 - x}$

Step 1

Take the log of both sides of the equation.

$\ln (7^{x^{2}}) = \ln (8^{9 - x})$

Step 2

Expand $\ln (7^{x^{2}})$ by moving $x^{2}$ outside the logarithm.

$x^{2} \ln (7) = \ln (8^{9 - x})$

Step 3

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

$x^{2} \ln (7) = (9 - x) \ln (8)$

Step 4

Solve the equation for $x$ .

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Step 4.1

Apply the distributive property.

$x^{2} \ln (7) = 9 \ln (8) - x \ln (8)$

Step 4.2

Add $x \ln (8)$ to both sides of the equation.

$x^{2} \ln (7) + x \ln (8) = 9 \ln (8)$

Step 4.3

Subtract $9 \ln (8)$ from both sides of the equation.

$x^{2} \ln (7) + x \ln (8) - 9 \ln (8) = 0$

Step 4.4

Use the quadratic formula to find the solutions.

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

Step 4.5

Substitute the values $a = \ln (7)$ , $b = \ln (8)$ , and $c = - 9 \ln (8)$ into the quadratic formula and solve for $x$ .

$\frac{- \ln (8) \pm \sqrt{\ln^{2} (8) - 4 \cdot (\ln (7) \cdot (- 9 \ln (8)))}}{2 \ln (7)}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Factor $\ln (8)$ out of $\ln^{2} (8) - 4 \cdot \ln (7) \cdot - 9 \ln (8)$ .

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Step 4.6.1.1

Factor $\ln (8)$ out of $\ln^{2} (8)$ .

$x = \frac{- \ln (8) \pm \sqrt{\ln (8) \ln (8) - 4 \cdot \ln (7) \cdot (- 9 \ln (8))}}{2 \ln (7)}$

Step 4.6.1.2

Factor $\ln (8)$ out of $- 4 \cdot \ln (7) \cdot - 9 \ln (8)$ .

$x = \frac{- \ln (8) \pm \sqrt{\ln (8) \ln (8) + \ln (8) (- 4 \cdot \ln (7) \cdot - 9)}}{2 \ln (7)}$

Step 4.6.1.3

Factor $\ln (8)$ out of $\ln (8) \ln (8) + \ln (8) (- 4 \cdot \ln (7) \cdot - 9)$ .

$x = \frac{- \ln (8) \pm \sqrt{\ln (8) (\ln (8) - 4 \cdot \ln (7) \cdot - 9)}}{2 \ln (7)}$

Step 4.6.2

Multiply $- 9$ by $- 4$ .

$x = \frac{- \ln (8) \pm \sqrt{\ln (8) (\ln (8) + 36 \ln (7))}}{2 \ln (7)}$

Step 4.7

The final answer is the combination of both solutions.

$x = - \frac{\ln (8) - \sqrt{\ln (8) (\ln (8) + 36 \ln (7))}}{2 \ln (7)}, - \frac{\ln (8) + \sqrt{\ln (8) (\ln (8) + 36 \ln (7))}}{2 \ln (7)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (8) - \sqrt{\ln (8) (\ln (8) + 36 \ln (7))}}{2 \ln (7)}, - \frac{\ln (8) + \sqrt{\ln (8) (\ln (8) + 36 \ln (7))}}{2 \ln (7)}$

Decimal Form:

$x = 2.61260549 \dots, - 3.68122705 \dots$