Trigonometry Examples

$\frac{1}{2} \cdot \log (x - 2) = - 2$

Step 1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot \log (x - 2)) = 2 \cdot - 2$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $2 (\frac{1}{2} \cdot \log (x - 2))$ .

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Step 2.1.1.1

Combine $\frac{1}{2}$ and $\log (x - 2)$ .

$2 \frac{\log (x - 2)}{2} = 2 \cdot - 2$

Step 2.1.1.2

Cancel the common factor of $2$ .

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Step 2.1.1.2.1

Cancel the common factor.

$2 \frac{\log (x - 2)}{2} = 2 \cdot - 2$

Step 2.1.1.2.2

Rewrite the expression.

$\log (x - 2) = 2 \cdot - 2$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $2$ by $- 2$ .

$\log (x - 2) = - 4$

Step 3

Rewrite $\log (x - 2) = - 4$ 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 $b^{y} = x$ .

$10^{- 4} = x - 2$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $x - 2 = 10^{- 4}$ .

$x - 2 = 10^{- 4}$

Step 4.2

Simplify $10^{- 4}$ .

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Step 4.2.1

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

$x - 2 = \frac{1}{10^{4}}$

Step 4.2.2

Raise $10$ to the power of $4$ .

$x - 2 = \frac{1}{10000}$

Step 4.3

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

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Step 4.3.1

Add $2$ to both sides of the equation.

$x = \frac{1}{10000} + 2$

Step 4.3.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{10000}{10000}$ .

$x = \frac{1}{10000} + 2 \cdot \frac{10000}{10000}$

Step 4.3.3

Combine $2$ and $\frac{10000}{10000}$ .

$x = \frac{1}{10000} + \frac{2 \cdot 10000}{10000}$

Step 4.3.4

Combine the numerators over the common denominator.

$x = \frac{1 + 2 \cdot 10000}{10000}$

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply $2$ by $10000$ .

$x = \frac{1 + 20000}{10000}$

Step 4.3.5.2

Add $1$ and $20000$ .

$x = \frac{20001}{10000}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{20001}{10000}$

Decimal Form:

$x = 2.0001$

Mixed Number Form:

$x = 2 \frac{1}{10000}$