Trigonometry Examples

$7^{y + 2} = 2$

Step 1

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

$\ln (7^{y + 2}) = \ln (2)$

Step 2

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

$(y + 2) \ln (7) = \ln (2)$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$y \ln (7) + 2 \ln (7) = \ln (2)$

Step 4

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

$y \ln (7) + 2 \ln (7) - \ln (2) = 0$

Step 5

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

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Step 5.1

Subtract $2 \ln (7)$ from both sides of the equation.

$y \ln (7) - \ln (2) = - 2 \ln (7)$

Step 5.2

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

$y \ln (7) = - 2 \ln (7) + \ln (2)$

Step 6

Divide each term in $y \ln (7) = - 2 \ln (7) + \ln (2)$ by $\ln (7)$ and simplify.

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Step 6.1

Divide each term in $y \ln (7) = - 2 \ln (7) + \ln (2)$ by $\ln (7)$ .

$\frac{y \ln (7)}{\ln (7)} = \frac{- 2 \ln (7)}{\ln (7)} + \frac{\ln (2)}{\ln (7)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $\ln (7)$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{y \ln (7)}{\ln (7)} = \frac{- 2 \ln (7)}{\ln (7)} + \frac{\ln (2)}{\ln (7)}$

Step 6.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 2 \ln (7)}{\ln (7)} + \frac{\ln (2)}{\ln (7)}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Cancel the common factor of $\ln (7)$ .

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Step 6.3.1.1

Cancel the common factor.

$y = \frac{- 2 \ln (7)}{\ln (7)} + \frac{\ln (2)}{\ln (7)}$

Step 6.3.1.2

Divide $- 2$ by $1$ .

$y = - 2 + \frac{\ln (2)}{\ln (7)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = - 2 + \frac{\ln (2)}{\ln (7)}$

Decimal Form:

$y = - 1.64379281 \dots$