Trigonometry Examples

$6^{x^{2}} = 28$

Step 1

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

$\ln (6^{x^{2}}) = \ln (28)$

Step 2

Expand $\ln (6^{x^{2}})$ by moving $x^{2}$ outside the logarithm.

$x^{2} \ln (6) = \ln (28)$

Step 3

Divide each term in $x^{2} \ln (6) = \ln (28)$ by $\ln (6)$ and simplify.

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Step 3.1

Divide each term in $x^{2} \ln (6) = \ln (28)$ by $\ln (6)$ .

$\frac{x^{2} \ln (6)}{\ln (6)} = \frac{\ln (28)}{\ln (6)}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $\ln (6)$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{x^{2} \ln (6)}{\ln (6)} = \frac{\ln (28)}{\ln (6)}$

Step 3.2.1.2

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

$x^{2} = \frac{\ln (28)}{\ln (6)}$

Step 4

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

$x = \pm \sqrt{\frac{\ln (28)}{\ln (6)}}$

Step 5

Simplify $\pm \sqrt{\frac{\ln (28)}{\ln (6)}}$ .

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Step 5.1

Rewrite $\sqrt{\frac{\ln (28)}{\ln (6)}}$ as $\frac{\sqrt{\ln (28)}}{\sqrt{\ln (6)}}$ .

$x = \pm \frac{\sqrt{\ln (28)}}{\sqrt{\ln (6)}}$

Step 5.2

Multiply $\frac{\sqrt{\ln (28)}}{\sqrt{\ln (6)}}$ by $\frac{\sqrt{\ln (6)}}{\sqrt{\ln (6)}}$ .

$x = \pm \frac{\sqrt{\ln (28)}}{\sqrt{\ln (6)}} \cdot \frac{\sqrt{\ln (6)}}{\sqrt{\ln (6)}}$

Step 5.3

Combine and simplify the denominator.

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Step 5.3.1

Multiply $\frac{\sqrt{\ln (28)}}{\sqrt{\ln (6)}}$ by $\frac{\sqrt{\ln (6)}}{\sqrt{\ln (6)}}$ .

$x = \pm \frac{\sqrt{\ln (28)} \sqrt{\ln (6)}}{\sqrt{\ln (6)} \sqrt{\ln (6)}}$

Step 5.3.2

Raise $\sqrt{\ln (6)}$ to the power of $1$ .

$x = \pm \frac{\sqrt{\ln (28)} \sqrt{\ln (6)}}{{\sqrt{\ln (6)}}^{1} \sqrt{\ln (6)}}$

Step 5.3.3

Raise $\sqrt{\ln (6)}$ to the power of $1$ .

$x = \pm \frac{\sqrt{\ln (28)} \sqrt{\ln (6)}}{{\sqrt{\ln (6)}}^{1} {\sqrt{\ln (6)}}^{1}}$

Step 5.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{\ln (28)} \sqrt{\ln (6)}}{{\sqrt{\ln (6)}}^{1 + 1}}$

Step 5.3.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{\ln (28)} \sqrt{\ln (6)}}{{\sqrt{\ln (6)}}^{2}}$

Step 5.3.6

Rewrite ${\sqrt{\ln (6)}}^{2}$ as $\ln (6)$ .

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Step 5.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\ln (6)}$ as ${\ln (6)}^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{\ln (28)} \sqrt{\ln (6)}}{{({\ln (6)}^{\frac{1}{2}})}^{2}}$

Step 5.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{\ln (28)} \sqrt{\ln (6)}}{{\ln (6)}^{\frac{1}{2} \cdot 2}}$

Step 5.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{\ln (28)} \sqrt{\ln (6)}}{{\ln (6)}^{\frac{2}{2}}}$

Step 5.3.6.4

Cancel the common factor of $2$ .

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Step 5.3.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{\ln (28)} \sqrt{\ln (6)}}{{\ln (6)}^{\frac{2}{2}}}$

Step 5.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{\ln (28)} \sqrt{\ln (6)}}{\ln^{1} (6)}$

Step 5.3.6.5

Simplify.

$x = \pm \frac{\sqrt{\ln (28)} \sqrt{\ln (6)}}{\ln (6)}$

Step 5.4

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{\ln (28) \ln (6)}}{\ln (6)}$

Step 6

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

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Step 6.1

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

$x = \frac{\sqrt{\ln (28) \ln (6)}}{\ln (6)}$

Step 6.2

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

$x = - \frac{\sqrt{\ln (28) \ln (6)}}{\ln (6)}$

Step 6.3

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

$x = \frac{\sqrt{\ln (28) \ln (6)}}{\ln (6)}, - \frac{\sqrt{\ln (28) \ln (6)}}{\ln (6)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{\ln (28) \ln (6)}}{\ln (6)}, - \frac{\sqrt{\ln (28) \ln (6)}}{\ln (6)}$

Decimal Form:

$x = 1.36372238 \dots, - 1.36372238 \dots$