Precalculus Examples

$2 = \log_{x + 1} (y + 1)$

Step 1

Rewrite the equation as $\log_{x + 1} (y + 1) = 2$ .

$\log_{x + 1} (y + 1) = 2$

Step 2

Rewrite $\log_{x + 1} (y + 1) = 2$ 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 $b^{y} = x$ .

${(x + 1)}^{2} = y + 1$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $y + 1 = {(x + 1)}^{2}$ .

$y + 1 = {(x + 1)}^{2}$

Step 3.2

Simplify ${(x + 1)}^{2}$ .

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Step 3.2.1

Rewrite.

$y + 1 = 0 + 0 + {(x + 1)}^{2}$

Step 3.2.2

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$y + 1 = (x + 1) (x + 1)$

Step 3.2.3

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 3.2.3.1

Apply the distributive property.

$y + 1 = x (x + 1) + 1 (x + 1)$

Step 3.2.3.2

Apply the distributive property.

$y + 1 = x \cdot x + x \cdot 1 + 1 (x + 1)$

Step 3.2.3.3

Apply the distributive property.

$y + 1 = x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1$

Step 3.2.4

Simplify and combine like terms.

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Step 3.2.4.1

Simplify each term.

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Step 3.2.4.1.1

Multiply $x$ by $x$ .

$y + 1 = x^{2} + x \cdot 1 + 1 x + 1 \cdot 1$

Step 3.2.4.1.2

Multiply $x$ by $1$ .

$y + 1 = x^{2} + x + 1 x + 1 \cdot 1$

Step 3.2.4.1.3

Multiply $x$ by $1$ .

$y + 1 = x^{2} + x + x + 1 \cdot 1$

Step 3.2.4.1.4

Multiply $1$ by $1$ .

$y + 1 = x^{2} + x + x + 1$

Step 3.2.4.2

Add $x$ and $x$ .

$y + 1 = x^{2} + 2 x + 1$

Step 3.3

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

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Step 3.3.1

Subtract $1$ from both sides of the equation.

$y = x^{2} + 2 x + 1 - 1$

Step 3.3.2

Combine the opposite terms in $x^{2} + 2 x + 1 - 1$ .

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Step 3.3.2.1

Subtract $1$ from $1$ .

$y = x^{2} + 2 x + 0$

Step 3.3.2.2

Add $x^{2} + 2 x$ and $0$ .

$y = x^{2} + 2 x$