Trigonometry Examples

$\ln (2 x - 2) = - 6$

Step 1

To solve for $x$ , rewrite the equation using properties of logarithms.

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

Step 2

Rewrite $\ln (2 x - 2) = - 6$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{- 6} = 2 x - 2$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $2 x - 2 = e^{- 6}$ .

$2 x - 2 = e^{- 6}$

Step 3.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 x - 2 = \frac{1}{e^{6}}$

Step 3.3

Add $2$ to both sides of the equation.

$2 x = \frac{1}{e^{6}} + 2$

Step 3.4

Divide each term in $2 x = \frac{1}{e^{6}} + 2$ by $2$ and simplify.

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Step 3.4.1

Divide each term in $2 x = \frac{1}{e^{6}} + 2$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{1}{e^{6}}}{2} + \frac{2}{2}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{1}{e^{6}}}{2} + \frac{2}{2}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{1}{e^{6}}}{2} + \frac{2}{2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify each term.

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Step 3.4.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{1}{e^{6}} \cdot \frac{1}{2} + \frac{2}{2}$

Step 3.4.3.1.2

Combine.

$x = \frac{1 \cdot 1}{e^{6} \cdot 2} + \frac{2}{2}$

Step 3.4.3.1.3

Multiply $1$ by $1$ .

$x = \frac{1}{e^{6} \cdot 2} + \frac{2}{2}$

Step 3.4.3.1.4

Move $2$ to the left of $e^{6}$ .

$x = \frac{1}{2 \cdot e^{6}} + \frac{2}{2}$

Step 3.4.3.1.5

Divide $2$ by $2$ .

$x = \frac{1}{2 e^{6}} + 1$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2 e^{6}} + 1$

Decimal Form:

$x = 1.00123937 \dots$