Trigonometry Examples

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

Step 1

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

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

Step 2

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

$(2 x + 4) \ln (2) = \ln (3)$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$2 x \ln (2) + 4 \ln (2) = \ln (3)$

Step 4

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

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

Step 5

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

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Step 5.1

Subtract $4 \ln (2)$ from both sides of the equation.

$2 x \ln (2) - \ln (3) = - 4 \ln (2)$

Step 5.2

Add $\ln (3)$ to both sides of the equation.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Rewrite the expression.

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

Step 6.2.2

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

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Step 6.2.2.1

Cancel the common factor.

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

Step 6.2.2.2

Divide $x$ by $1$ .

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

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Cancel the common factor of $- 4$ and $2$ .

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Step 6.3.1.1.1

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

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

Step 6.3.1.1.2

Cancel the common factors.

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Step 6.3.1.1.2.1

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

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

Step 6.3.1.1.2.2

Cancel the common factor.

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

Step 6.3.1.1.2.3

Rewrite the expression.

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

Step 6.3.1.2

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

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Step 6.3.1.2.1

Cancel the common factor.

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

Step 6.3.1.2.2

Divide $- 2$ by $1$ .

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 1.20751874 \dots$

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