Algebra Examples

$2^{x^{2} - 4} = 3^{x^{2} - 4}$

Step 1

Take the log of both sides of the equation.

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

Step 2

Expand $\ln (2^{x^{2} - 4})$ by moving $x^{2} - 4$ outside the logarithm.

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

Step 3

Expand $\ln (3^{x^{2} - 4})$ by moving $x^{2} - 4$ outside the logarithm.

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

Step 4

Solve the equation for $x$ .

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Step 4.1

Simplify $(x^{2} - 4) \ln (2)$ .

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Step 4.1.1

Rewrite.

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

Step 4.1.2

Simplify by adding zeros.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Subtract $x^{2} \ln (3)$ from both sides of the equation.

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

Step 4.4

Add $4 \ln (2)$ to both sides of the equation.

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

Step 4.5

Factor $x^{2}$ out of $x^{2} \ln (2) - x^{2} \ln (3)$ .

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Step 4.5.1

Factor $x^{2}$ out of $- x^{2} \ln (3)$ .

$x^{2} \ln (2) + x^{2} (- 1 \ln (3)) = - 4 \ln (3) + 4 \ln (2)$

Step 4.5.2

Factor $x^{2}$ out of $x^{2} \ln (2) + x^{2} (- 1 \ln (3))$ .

$x^{2} (\ln (2) - 1 \ln (3)) = - 4 \ln (3) + 4 \ln (2)$

Step 4.6

Rewrite $- 1 \ln (3)$ as $- \ln (3)$ .

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

Step 4.7

Divide each term in $x^{2} (\ln (2) - \ln (3)) = - 4 \ln (3) + 4 \ln (2)$ by $\ln (2) - \ln (3)$ and simplify.

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Step 4.7.1

Divide each term in $x^{2} (\ln (2) - \ln (3)) = - 4 \ln (3) + 4 \ln (2)$ by $\ln (2) - \ln (3)$ .

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

Step 4.7.2

Simplify the left side.

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Step 4.7.2.1

Cancel the common factor of $\ln (2) - \ln (3)$ .

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Step 4.7.2.1.1

Cancel the common factor.

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

Step 4.7.2.1.2

Divide $x^{2}$ by $1$ .

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

Step 4.7.3

Simplify the right side.

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Step 4.7.3.1

Combine the numerators over the common denominator.

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

Step 4.7.3.2

Factor $4$ out of $- 4 \ln (3) + 4 \ln (2)$ .

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Step 4.7.3.2.1

Factor $4$ out of $- 4 \ln (3)$ .

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

Step 4.7.3.2.2

Factor $4$ out of $4 \ln (2)$ .

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

Step 4.7.3.2.3

Factor $4$ out of $4 (- \ln (3)) + 4 \ln (2)$ .

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

Step 4.7.3.3

Cancel the common factor of $- \ln (3) + \ln (2)$ and $\ln (2) - \ln (3)$ .

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Step 4.7.3.3.1

Reorder terms.

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

Step 4.7.3.3.2

Cancel the common factor.

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

Step 4.7.3.3.3

Divide $4$ by $1$ .

$x^{2} = 4$

Step 4.8

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4}$

Step 4.9

Simplify $\pm \sqrt{4}$ .

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Step 4.9.1

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

$x = \pm \sqrt{2^{2}}$

Step 4.9.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

Step 4.10

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.10.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Step 4.10.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

Step 4.10.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$