Precalculus Examples

$\ln (2 x + 5) + \ln (x - 7) - 2 \ln (x) = 0$

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln ((2 x + 5) (x - 7)) - 2 \ln (x) = 0$

Step 1.2

Simplify each term.

Tap for more steps...

Step 1.2.1

Expand $(2 x + 5) (x - 7)$ using the FOIL Method.

Tap for more steps...

Step 1.2.1.1

Apply the distributive property.

$\ln (2 x (x - 7) + 5 (x - 7)) - 2 \ln (x) = 0$

Step 1.2.1.2

Apply the distributive property.

$\ln (2 x \cdot x + 2 x \cdot - 7 + 5 (x - 7)) - 2 \ln (x) = 0$

Step 1.2.1.3

Apply the distributive property.

$\ln (2 x \cdot x + 2 x \cdot - 7 + 5 x + 5 \cdot - 7) - 2 \ln (x) = 0$

Step 1.2.2

Simplify and combine like terms.

Tap for more steps...

Step 1.2.2.1

Simplify each term.

Tap for more steps...

Step 1.2.2.1.1

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.2.2.1.1.1

Move $x$ .

$\ln (2 (x \cdot x) + 2 x \cdot - 7 + 5 x + 5 \cdot - 7) - 2 \ln (x) = 0$

Step 1.2.2.1.1.2

Multiply $x$ by $x$ .

$\ln (2 x^{2} + 2 x \cdot - 7 + 5 x + 5 \cdot - 7) - 2 \ln (x) = 0$

Step 1.2.2.1.2

Multiply $- 7$ by $2$ .

$\ln (2 x^{2} - 14 x + 5 x + 5 \cdot - 7) - 2 \ln (x) = 0$

Step 1.2.2.1.3

Multiply $5$ by $- 7$ .

$\ln (2 x^{2} - 14 x + 5 x - 35) - 2 \ln (x) = 0$

Step 1.2.2.2

Add $- 14 x$ and $5 x$ .

$\ln (2 x^{2} - 9 x - 35) - 2 \ln (x) = 0$

Step 2

Simplify the left side.

Tap for more steps...

Step 2.1

Simplify $\ln (2 x^{2} - 9 x - 35) - 2 \ln (x)$ .

Tap for more steps...

Step 2.1.1

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

$\ln (2 x^{2} - 9 x - 35) - \ln (x^{2}) = 0$

Step 2.1.2

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

$\ln (\frac{2 x^{2} - 9 x - 35}{x^{2}}) = 0$

Step 2.1.3

Factor by grouping.

Tap for more steps...

Step 2.1.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 35 = - 70$ and whose sum is $b = - 9$ .

Tap for more steps...

Step 2.1.3.1.1

Factor $- 9$ out of $- 9 x$ .

$\ln (\frac{2 x^{2} - 9 (x) - 35}{x^{2}}) = 0$

Step 2.1.3.1.2

Rewrite $- 9$ as $5$ plus $- 14$

$\ln (\frac{2 x^{2} + (5 - 14) x - 35}{x^{2}}) = 0$

Step 2.1.3.1.3

Apply the distributive property.

$\ln (\frac{2 x^{2} + 5 x - 14 x - 35}{x^{2}}) = 0$

Step 2.1.3.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.1.3.2.1

Group the first two terms and the last two terms.

$\ln (\frac{(2 x^{2} + 5 x) - 14 x - 35}{x^{2}}) = 0$

Step 2.1.3.2.2

Factor out the greatest common factor (GCF) from each group.

$\ln (\frac{x (2 x + 5) - 7 (2 x + 5)}{x^{2}}) = 0$

Step 2.1.3.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 5$ .

$\ln (\frac{(2 x + 5) (x - 7)}{x^{2}}) = 0$

Step 3

Rewrite $\ln (\frac{(2 x + 5) (x - 7)}{x^{2}}) = 0$ 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 $b^{y} = x$ .

$e^{0} = \frac{(2 x + 5) (x - 7)}{x^{2}}$

Step 4

Cross multiply to remove the fraction.

$(2 x + 5) (x - 7) = e^{0} (x^{2})$

Step 5

Simplify $e^{0} (x^{2})$ .

Tap for more steps...

Step 5.1

Anything raised to $0$ is $1$ .

$(2 x + 5) (x - 7) = 1 x^{2}$

Step 5.2

Multiply $x^{2}$ by $1$ .

$(2 x + 5) (x - 7) = x^{2}$

Step 6

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

Tap for more steps...

Step 6.1

Subtract $x^{2}$ from both sides of the equation.

$(2 x + 5) (x - 7) - x^{2} = 0$

Step 6.2

Simplify each term.

Tap for more steps...

Step 6.2.1

Expand $(2 x + 5) (x - 7)$ using the FOIL Method.

Tap for more steps...

Step 6.2.1.1

Apply the distributive property.

$2 x (x - 7) + 5 (x - 7) - x^{2} = 0$

Step 6.2.1.2

Apply the distributive property.

$2 x \cdot x + 2 x \cdot - 7 + 5 (x - 7) - x^{2} = 0$

Step 6.2.1.3

Apply the distributive property.

$2 x \cdot x + 2 x \cdot - 7 + 5 x + 5 \cdot - 7 - x^{2} = 0$

Step 6.2.2

Simplify and combine like terms.

Tap for more steps...

Step 6.2.2.1

Simplify each term.

Tap for more steps...

Step 6.2.2.1.1

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 6.2.2.1.1.1

Move $x$ .

$2 (x \cdot x) + 2 x \cdot - 7 + 5 x + 5 \cdot - 7 - x^{2} = 0$

Step 6.2.2.1.1.2

Multiply $x$ by $x$ .

$2 x^{2} + 2 x \cdot - 7 + 5 x + 5 \cdot - 7 - x^{2} = 0$

Step 6.2.2.1.2

Multiply $- 7$ by $2$ .

$2 x^{2} - 14 x + 5 x + 5 \cdot - 7 - x^{2} = 0$

Step 6.2.2.1.3

Multiply $5$ by $- 7$ .

$2 x^{2} - 14 x + 5 x - 35 - x^{2} = 0$

Step 6.2.2.2

Add $- 14 x$ and $5 x$ .

$2 x^{2} - 9 x - 35 - x^{2} = 0$

Step 6.3

Subtract $x^{2}$ from $2 x^{2}$ .

$x^{2} - 9 x - 35 = 0$

Step 7

Add $35$ to both sides of the equation.

$x^{2} - 9 x = 35$

Step 8

Subtract $35$ from both sides of the equation.

$x^{2} - 9 x - 35 = 0$

Step 9

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 10

Substitute the values $a = 1$ , $b = - 9$ , and $c = - 35$ into the quadratic formula and solve for $x$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (1 \cdot - 35)}}{2 \cdot 1}$

Step 11

Simplify.

Tap for more steps...

Step 11.1

Simplify the numerator.

Tap for more steps...

Step 11.1.1

Raise $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot - 35}}{2 \cdot 1}$

Step 11.1.2

Multiply $- 4 \cdot 1 \cdot - 35$ .

Tap for more steps...

Step 11.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot - 35}}{2 \cdot 1}$

Step 11.1.2.2

Multiply $- 4$ by $- 35$ .

$x = \frac{9 \pm \sqrt{81 + 140}}{2 \cdot 1}$

Step 11.1.3

Add $81$ and $140$ .

$x = \frac{9 \pm \sqrt{221}}{2 \cdot 1}$

Step 11.2

Multiply $2$ by $1$ .

$x = \frac{9 \pm \sqrt{221}}{2}$

Step 12

The final answer is the combination of both solutions.

$x = \frac{9 + \sqrt{221}}{2}, \frac{9 - \sqrt{221}}{2}$

Step 13

Exclude the solutions that do not make $\ln (2 x + 5) + \ln (x - 7) - 2 \ln (x) = 0$ true.

$x = \frac{9 + \sqrt{221}}{2}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$x = \frac{9 + \sqrt{221}}{2}$

Decimal Form:

$x = 11.93303437 \dots$