Finite Math Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Apply the distributive property.

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

Step 4

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

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Step 4.1

Move $x$ .

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

Step 4.2

Multiply $x$ by $x$ .

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

Step 5

Multiply $x$ by $1$ .

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

Step 6

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

$e^{\ln (2 x^{2} + x)} = e^{2}$

Step 7

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

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

Step 8

Solve for $x$ .

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Step 8.1

Rewrite the equation as $2 x^{2} + x = e^{2}$ .

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

Step 8.2

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

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

Step 8.3

Use the quadratic formula to find the solutions.

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

Step 8.4

Substitute the values $a = 2$ , $b = 1$ , and $c = - e^{2}$ into the quadratic formula and solve for $x$ .

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

Step 8.5

Simplify.

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Step 8.5.1

Simplify the numerator.

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Step 8.5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 2 \cdot (- 1 e^{2})}}{2 \cdot 2}$

Step 8.5.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 8.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 - 8 \cdot (- 1 e^{2})}}{2 \cdot 2}$

Step 8.5.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{1 + 8 e^{2}}}{2 \cdot 2}$

Step 8.5.2

Multiply $2$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 + 8 e^{2}}}{4}$

Step 8.6

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{1 + 8 e^{2}}}{4}, - \frac{1 + \sqrt{1 + 8 e^{2}}}{4}$

Step 9

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

$x = - \frac{1 - \sqrt{1 + 8 e^{2}}}{4}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 - \sqrt{1 + 8 e^{2}}}{4}$

Decimal Form:

$x = 1.68830545 \dots$