Precalculus Examples

$2 \ln (x + 3) - \ln (x + 1) = 3 \ln (2)$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$2 \ln (x + 3) - \ln (x + 1) - 3 \ln (2) = 0$

Step 2

Simplify the left side.

Tap for more steps...

Step 2.1

Simplify $2 \ln (x + 3) - \ln (x + 1) - 3 \ln (2)$ .

Tap for more steps...

Step 2.1.1

Simplify each term.

Tap for more steps...

Step 2.1.1.1

Simplify $2 \ln (x + 3)$ by moving $2$ inside the logarithm.

$\ln ({(x + 3)}^{2}) - \ln (x + 1) - 3 \ln (2) = 0$

Step 2.1.1.2

Simplify $- 3 \ln (2)$ by moving $3$ inside the logarithm.

$\ln ({(x + 3)}^{2}) - \ln (x + 1) - \ln (2^{3}) = 0$

Step 2.1.1.3

Raise $2$ to the power of $3$ .

$\ln ({(x + 3)}^{2}) - \ln (x + 1) - \ln (8) = 0$

Step 2.1.2

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

$\ln (\frac{{(x + 3)}^{2}}{x + 1}) - \ln (8) = 0$

Step 2.1.3

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

$\ln (\frac{\frac{{(x + 3)}^{2}}{x + 1}}{8}) = 0$

Step 2.1.4

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{{(x + 3)}^{2}}{x + 1} \cdot \frac{1}{8}) = 0$

Step 2.1.5

Multiply $\frac{{(x + 3)}^{2}}{x + 1}$ by $\frac{1}{8}$ .

$\ln (\frac{{(x + 3)}^{2}}{(x + 1) \cdot 8}) = 0$

Step 2.1.6

Move $8$ to the left of $x + 1$ .

$\ln (\frac{{(x + 3)}^{2}}{8 (x + 1)}) = 0$

Step 3

Rewrite $\ln (\frac{{(x + 3)}^{2}}{8 (x + 1)}) = 0$ 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 $b^{y} = x$ .

$e^{0} = \frac{{(x + 3)}^{2}}{8 (x + 1)}$

Step 4

Cross multiply to remove the fraction.

${(x + 3)}^{2} = e^{0} (8 (x + 1))$

Step 5

Simplify $e^{0} (8 (x + 1))$ .

Tap for more steps...

Step 5.1

Simplify the expression.

Tap for more steps...

Step 5.1.1

Anything raised to $0$ is $1$ .

${(x + 3)}^{2} = 1 (8 (x + 1))$

Step 5.1.2

Multiply $8 (x + 1)$ by $1$ .

${(x + 3)}^{2} = 8 (x + 1)$

Step 5.2

Apply the distributive property.

${(x + 3)}^{2} = 8 x + 8 \cdot 1$

Step 5.3

Multiply $8$ by $1$ .

${(x + 3)}^{2} = 8 x + 8$

Step 6

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

Tap for more steps...

Step 6.1

Subtract $8 x$ from both sides of the equation.

${(x + 3)}^{2} - 8 x = 8$

Step 6.2

Simplify each term.

Tap for more steps...

Step 6.2.1

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$(x + 3) (x + 3) - 8 x = 8$

Step 6.2.2

Expand $(x + 3) (x + 3)$ using the FOIL Method.

Tap for more steps...

Step 6.2.2.1

Apply the distributive property.

$x (x + 3) + 3 (x + 3) - 8 x = 8$

Step 6.2.2.2

Apply the distributive property.

$x \cdot x + x \cdot 3 + 3 (x + 3) - 8 x = 8$

Step 6.2.2.3

Apply the distributive property.

$x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3 - 8 x = 8$

Step 6.2.3

Simplify and combine like terms.

Tap for more steps...

Step 6.2.3.1

Simplify each term.

Tap for more steps...

Step 6.2.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 3 + 3 x + 3 \cdot 3 - 8 x = 8$

Step 6.2.3.1.2

Move $3$ to the left of $x$ .

$x^{2} + 3 \cdot x + 3 x + 3 \cdot 3 - 8 x = 8$

Step 6.2.3.1.3

Multiply $3$ by $3$ .

$x^{2} + 3 x + 3 x + 9 - 8 x = 8$

Step 6.2.3.2

Add $3 x$ and $3 x$ .

$x^{2} + 6 x + 9 - 8 x = 8$

Step 6.3

Subtract $8 x$ from $6 x$ .

$x^{2} - 2 x + 9 = 8$

Step 7

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

Tap for more steps...

Step 7.1

Subtract $9$ from both sides of the equation.

$x^{2} - 2 x = 8 - 9$

Step 7.2

Subtract $9$ from $8$ .

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

Step 8

Add $1$ to both sides of the equation.

$x^{2} - 2 x + 1 = 0$

Step 9

Factor using the perfect square rule.

Tap for more steps...

Step 9.1

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

$x^{2} - 2 x + 1^{2} = 0$

Step 9.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 9.3

Rewrite the polynomial.

$x^{2} - 2 \cdot x \cdot 1 + 1^{2} = 0$

Step 9.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

${(x - 1)}^{2} = 0$

Step 10

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 11

Add $1$ to both sides of the equation.

$x = 1$