Trigonometry Examples

$4026^{2 x + 4} = 1024^{- 4 x + 2}$

Step 1

Take the log of both sides of the equation.

$\ln (4026^{2 x + 4}) = \ln (1024^{- 4 x + 2})$

Step 2

Expand $\ln (4026^{2 x + 4})$ by moving $2 x + 4$ outside the logarithm.

$(2 x + 4) \ln (4026) = \ln (1024^{- 4 x + 2})$

Step 3

Expand $\ln (1024^{- 4 x + 2})$ by moving $- 4 x + 2$ outside the logarithm.

$(2 x + 4) \ln (4026) = (- 4 x + 2) \ln (1024)$

Step 4

Solve the equation for $x$ .

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Step 4.1

Simplify $(2 x + 4) \ln (4026)$ .

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Step 4.1.1

Rewrite.

$0 + 0 + (2 x + 4) \ln (4026) = (- 4 x + 2) \ln (1024)$

Step 4.1.2

Simplify by adding zeros.

$(2 x + 4) \ln (4026) = (- 4 x + 2) \ln (1024)$

Step 4.1.3

Apply the distributive property.

$2 x \ln (4026) + 4 \ln (4026) = (- 4 x + 2) \ln (1024)$

Step 4.2

Apply the distributive property.

$2 x \ln (4026) + 4 \ln (4026) = - 4 x \ln (1024) + 2 \ln (1024)$

Step 4.3

Add $4 x \ln (1024)$ to both sides of the equation.

$2 x \ln (4026) + 4 \ln (4026) + 4 x \ln (1024) = 2 \ln (1024)$

Step 4.4

Subtract $4 \ln (4026)$ from both sides of the equation.

$2 x \ln (4026) + 4 x \ln (1024) = 2 \ln (1024) - 4 \ln (4026)$

Step 4.5

Factor $2 x$ out of $2 x \ln (4026) + 4 x \ln (1024)$ .

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Step 4.5.1

Factor $2 x$ out of $4 x \ln (1024)$ .

$2 x \ln (4026) + 2 x (2 \ln (1024)) = 2 \ln (1024) - 4 \ln (4026)$

Step 4.5.2

Factor $2 x$ out of $2 x \ln (4026) + 2 x (2 \ln (1024))$ .

$2 x (\ln (4026) + 2 \ln (1024)) = 2 \ln (1024) - 4 \ln (4026)$

Step 4.6

Divide each term in $2 x (\ln (4026) + 2 \ln (1024)) = 2 \ln (1024) - 4 \ln (4026)$ by $2 (\ln (4026) + 2 \ln (1024))$ and simplify.

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Step 4.6.1

Divide each term in $2 x (\ln (4026) + 2 \ln (1024)) = 2 \ln (1024) - 4 \ln (4026)$ by $2 (\ln (4026) + 2 \ln (1024))$ .

$\frac{2 x (\ln (4026) + 2 \ln (1024))}{2 (\ln (4026) + 2 \ln (1024))} = \frac{2 \ln (1024)}{2 (\ln (4026) + 2 \ln (1024))} + \frac{- 4 \ln (4026)}{2 (\ln (4026) + 2 \ln (1024))}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $2$ .

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Step 4.6.2.1.1

Cancel the common factor.

$\frac{2 x (\ln (4026) + 2 \ln (1024))}{2 (\ln (4026) + 2 \ln (1024))} = \frac{2 \ln (1024)}{2 (\ln (4026) + 2 \ln (1024))} + \frac{- 4 \ln (4026)}{2 (\ln (4026) + 2 \ln (1024))}$

Step 4.6.2.1.2

Rewrite the expression.

$\frac{x (\ln (4026) + 2 \ln (1024))}{\ln (4026) + 2 \ln (1024)} = \frac{2 \ln (1024)}{2 (\ln (4026) + 2 \ln (1024))} + \frac{- 4 \ln (4026)}{2 (\ln (4026) + 2 \ln (1024))}$

Step 4.6.2.2

Cancel the common factor of $\ln (4026) + 2 \ln (1024)$ .

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Step 4.6.2.2.1

Cancel the common factor.

$\frac{x (\ln (4026) + 2 \ln (1024))}{\ln (4026) + 2 \ln (1024)} = \frac{2 \ln (1024)}{2 (\ln (4026) + 2 \ln (1024))} + \frac{- 4 \ln (4026)}{2 (\ln (4026) + 2 \ln (1024))}$

Step 4.6.2.2.2

Divide $x$ by $1$ .

$x = \frac{2 \ln (1024)}{2 (\ln (4026) + 2 \ln (1024))} + \frac{- 4 \ln (4026)}{2 (\ln (4026) + 2 \ln (1024))}$

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Simplify each term.

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Step 4.6.3.1.1

Cancel the common factor of $2$ .

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Step 4.6.3.1.1.1

Cancel the common factor.

$x = \frac{2 \ln (1024)}{2 (\ln (4026) + 2 \ln (1024))} + \frac{- 4 \ln (4026)}{2 (\ln (4026) + 2 \ln (1024))}$

Step 4.6.3.1.1.2

Rewrite the expression.

$x = \frac{\ln (1024)}{\ln (4026) + 2 \ln (1024)} + \frac{- 4 \ln (4026)}{2 (\ln (4026) + 2 \ln (1024))}$

Step 4.6.3.1.2

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

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Step 4.6.3.1.2.1

Factor $2$ out of $- 4 \ln (4026)$ .

$x = \frac{\ln (1024)}{\ln (4026) + 2 \ln (1024)} + \frac{2 (- 2 \ln (4026))}{2 (\ln (4026) + 2 \ln (1024))}$

Step 4.6.3.1.2.2

Cancel the common factors.

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Step 4.6.3.1.2.2.1

Cancel the common factor.

$x = \frac{\ln (1024)}{\ln (4026) + 2 \ln (1024)} + \frac{2 (- 2 \ln (4026))}{2 (\ln (4026) + 2 \ln (1024))}$

Step 4.6.3.1.2.2.2

Rewrite the expression.

$x = \frac{\ln (1024)}{\ln (4026) + 2 \ln (1024)} + \frac{- 2 \ln (4026)}{\ln (4026) + 2 \ln (1024)}$

Step 4.6.3.1.3

Move the negative in front of the fraction.

$x = \frac{\ln (1024)}{\ln (4026) + 2 \ln (1024)} - \frac{2 \ln (4026)}{\ln (4026) + 2 \ln (1024)}$

Step 4.6.3.2

Combine the numerators over the common denominator.

$x = \frac{\ln (1024) - 2 \ln (4026)}{\ln (4026) + 2 \ln (1024)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (1024) - 2 \ln (4026)}{\ln (4026) + 2 \ln (1024)}$

Decimal Form:

$x = - 0.43628477 \dots$