Calculus Examples

$\ln (y^{2} - 1) - \ln (y + 1) = \ln (\sin (x))$

Step 1

Simplify the left side.

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Step 1.1

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

$\ln (\frac{y^{2} - 1}{y + 1}) = \ln (\sin (x))$

Step 1.2

Simplify the numerator.

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Step 1.2.1

Rewrite $1$ as $1^{2}$ .

$\ln (\frac{y^{2} - 1^{2}}{y + 1}) = \ln (\sin (x))$

Step 1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 1$ .

$\ln (\frac{(y + 1) (y - 1)}{y + 1}) = \ln (\sin (x))$

Step 1.3

Cancel the common factor of $y + 1$ .

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Step 1.3.1

Cancel the common factor.

$\ln (\frac{(y + 1) (y - 1)}{y + 1}) = \ln (\sin (x))$

Step 1.3.2

Divide $y - 1$ by $1$ .

$\ln (y - 1) = \ln (\sin (x))$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$y - 1 = \sin (x)$

Step 3

Add $1$ to both sides of the equation.

$y = \sin (x) + 1$