Calculus Examples

$9^{2 + y^{2}} = 225$

Step 1

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

$\ln (9^{2 + y^{2}}) = \ln (225)$

Step 2

Expand $\ln (9^{2 + y^{2}})$ by moving $2 + y^{2}$ outside the logarithm.

$(2 + y^{2}) \ln (9) = \ln (225)$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$2 \ln (9) + y^{2} \ln (9) = \ln (225)$

Step 4

Reorder $2 \ln (9)$ and $y^{2} \ln (9)$ .

$y^{2} \ln (9) + 2 \ln (9) = \ln (225)$

Step 5

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

$y^{2} \ln (9) + 2 \ln (9) - \ln (225) = 0$

Step 6

Move all terms not containing $y$ to the right side of the equation.

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Step 6.1

Subtract $2 \ln (9)$ from both sides of the equation.

$y^{2} \ln (9) - \ln (225) = - 2 \ln (9)$

Step 6.2

Add $\ln (225)$ to both sides of the equation.

$y^{2} \ln (9) = - 2 \ln (9) + \ln (225)$

Step 7

Divide each term in $y^{2} \ln (9) = - 2 \ln (9) + \ln (225)$ by $\ln (9)$ and simplify.

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Step 7.1

Divide each term in $y^{2} \ln (9) = - 2 \ln (9) + \ln (225)$ by $\ln (9)$ .

$\frac{y^{2} \ln (9)}{\ln (9)} = \frac{- 2 \ln (9)}{\ln (9)} + \frac{\ln (225)}{\ln (9)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $\ln (9)$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{y^{2} \ln (9)}{\ln (9)} = \frac{- 2 \ln (9)}{\ln (9)} + \frac{\ln (225)}{\ln (9)}$

Step 7.2.1.2

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

$y^{2} = \frac{- 2 \ln (9)}{\ln (9)} + \frac{\ln (225)}{\ln (9)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Cancel the common factor of $\ln (9)$ .

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Step 7.3.1.1

Cancel the common factor.

$y^{2} = \frac{- 2 \ln (9)}{\ln (9)} + \frac{\ln (225)}{\ln (9)}$

Step 7.3.1.2

Divide $- 2$ by $1$ .

$y^{2} = - 2 + \frac{\ln (225)}{\ln (9)}$

Step 8

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

$y = \pm \sqrt{- 2 + \frac{\ln (225)}{\ln (9)}}$

Step 9

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

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Step 9.1

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

$y = \sqrt{- 2 + \frac{\ln (225)}{\ln (9)}}$

Step 9.2

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

$y = - \sqrt{- 2 + \frac{\ln (225)}{\ln (9)}}$

Step 9.3

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

$y = \sqrt{- 2 + \frac{\ln (225)}{\ln (9)}}, - \sqrt{- 2 + \frac{\ln (225)}{\ln (9)}}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$y = \sqrt{- 2 + \frac{\ln (225)}{\ln (9)}}, - \sqrt{- 2 + \frac{\ln (225)}{\ln (9)}}$

Decimal Form:

$y = 0.68188966 \dots, - 0.68188966 \dots$