Finite Math Examples

$\ln (x - 2) + \ln (2 x + 1) = 8$

Step 1

Simplify the left side.

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Step 1.1

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

$\ln ((x - 2) (2 x + 1)) = 8$

Step 1.2

Expand $(x - 2) (2 x + 1)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\ln (x (2 x + 1) - 2 (2 x + 1)) = 8$

Step 1.2.2

Apply the distributive property.

$\ln (x (2 x) + x \cdot 1 - 2 (2 x + 1)) = 8$

Step 1.2.3

Apply the distributive property.

$\ln (x (2 x) + x \cdot 1 - 2 (2 x) - 2 \cdot 1) = 8$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite using the commutative property of multiplication.

$\ln (2 x \cdot x + x \cdot 1 - 2 (2 x) - 2 \cdot 1) = 8$

Step 1.3.1.2

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

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Step 1.3.1.2.1

Move $x$ .

$\ln (2 (x \cdot x) + x \cdot 1 - 2 (2 x) - 2 \cdot 1) = 8$

Step 1.3.1.2.2

Multiply $x$ by $x$ .

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

Step 1.3.1.3

Multiply $x$ by $1$ .

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

Step 1.3.1.4

Multiply $2$ by $- 2$ .

$\ln (2 x^{2} + x - 4 x - 2 \cdot 1) = 8$

Step 1.3.1.5

Multiply $- 2$ by $1$ .

$\ln (2 x^{2} + x - 4 x - 2) = 8$

Step 1.3.2

Subtract $4 x$ from $x$ .

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

Step 2

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (2 x^{2} - 3 x - 2)} = e^{8}$

Step 3

Rewrite $\ln (2 x^{2} - 3 x - 2) = 8$ 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 $b^{y} = x$ .

$e^{8} = 2 x^{2} - 3 x - 2$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $2 x^{2} - 3 x - 2 = e^{8}$ .

$2 x^{2} - 3 x - 2 = e^{8}$

Step 4.2

Subtract $e^{8}$ from both sides of the equation.

$2 x^{2} - 3 x - 2 - e^{8} = 0$

Step 4.3

Use the quadratic formula to find the solutions.

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

Step 4.4

Substitute the values $a = 2$ , $b = - 3$ , and $c = - 2 - e^{8}$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (2 \cdot (- 2 - e^{8}))}}{2 \cdot 2}$

Step 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 2 \cdot (- 2 - e^{8})}}{2 \cdot 2}$

Step 4.5.1.2

Multiply $- 4$ by $2$ .

$x = \frac{3 \pm \sqrt{9 - 8 \cdot (- 2 - e^{8})}}{2 \cdot 2}$

Step 4.5.1.3

Apply the distributive property.

$x = \frac{3 \pm \sqrt{9 - 8 \cdot - 2 - 8 (- e^{8})}}{2 \cdot 2}$

Step 4.5.1.4

Multiply $- 8$ by $- 2$ .

$x = \frac{3 \pm \sqrt{9 + 16 - 8 (- e^{8})}}{2 \cdot 2}$

Step 4.5.1.5

Multiply $- 1$ by $- 8$ .

$x = \frac{3 \pm \sqrt{9 + 16 + 8 e^{8}}}{2 \cdot 2}$

Step 4.5.1.6

Add $9$ and $16$ .

$x = \frac{3 \pm \sqrt{25 + 8 e^{8}}}{2 \cdot 2}$

Step 4.5.2

Multiply $2$ by $2$ .

$x = \frac{3 \pm \sqrt{25 + 8 e^{8}}}{4}$

Step 4.6

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{25 + 8 e^{8}}}{4}, \frac{3 - \sqrt{25 + 8 e^{8}}}{4}$

Step 5

Exclude the solutions that do not make $\ln (x - 2) + \ln (2 x + 1) = 8$ true.

$x = \frac{3 + \sqrt{25 + 8 e^{8}}}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + \sqrt{25 + 8 e^{8}}}{4}$

Decimal Form:

$x = 39.37695294 \dots$