Trigonometry Examples

$\log (y) - \log (2 - y) = 0$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{y}{2 - y}) = 0$

Step 2

Rewrite $\log (\frac{y}{2 - y}) = 0$ 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 $b^{y} = x$ .

$10^{0} = \frac{y}{2 - y}$

Step 3

Cross multiply to remove the fraction.

$y = 10^{0} (2 - y)$

Step 4

Simplify $10^{0} (2 - y)$ .

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Step 4.1

Anything raised to $0$ is $1$ .

$y = 1 (2 - y)$

Step 4.2

Multiply $2 - y$ by $1$ .

$y = 2 - y$

Step 5

Move all terms containing $y$ to the left side of the equation.

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Step 5.1

Add $y$ to both sides of the equation.

$y + y = 2$

Step 5.2

Add $y$ and $y$ .

$2 y = 2$

Step 6

Divide each term in $2 y = 2$ by $2$ and simplify.

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Step 6.1

Divide each term in $2 y = 2$ by $2$ .

$\frac{2 y}{2} = \frac{2}{2}$

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.

$\frac{2 y}{2} = \frac{2}{2}$

Step 6.2.1.2

Divide $y$ by $1$ .

$y = \frac{2}{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $2$ by $2$ .

$y = 1$