Finite Math Examples

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

Step 1

Factor the left side of the equation.

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Step 1.1

Let $u = \ln (x)$ . Substitute $u$ for all occurrences of $\ln (x)$ .

$u^{2} - u - 2 = 0$

Step 1.2

Factor $u^{2} - u - 2$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 1.2.2

Write the factored form using these integers.

$(u - 2) (u + 1) = 0$

Step 1.3

Replace all occurrences of $u$ with $\ln (x)$ .

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

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\ln (x) - 2 = 0$

$\ln (x) + 1 = 0$

Step 3

Set $\ln (x) - 2$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $\ln (x) - 2$ equal to $0$ .

$\ln (x) - 2 = 0$

Step 3.2

Solve $\ln (x) - 2 = 0$ for $x$ .

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Step 3.2.1

Add $2$ to both sides of the equation.

$\ln (x) = 2$

Step 3.2.2

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

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

Step 3.2.3

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

$e^{2} = x$

Step 3.2.4

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

$x = e^{2}$

Step 4

Set $\ln (x) + 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $\ln (x) + 1$ equal to $0$ .

$\ln (x) + 1 = 0$

Step 4.2

Solve $\ln (x) + 1 = 0$ for $x$ .

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$\ln (x) = - 1$

Step 4.2.2

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

$e^{\ln (x)} = e^{- 1}$

Step 4.2.3

Rewrite $\ln (x) = - 1$ 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 $b^{y} = x$ .

$e^{- 1} = x$

Step 4.2.4

Solve for $x$ .

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Step 4.2.4.1

Rewrite the equation as $x = e^{- 1}$ .

$x = e^{- 1}$

Step 4.2.4.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x = \frac{1}{e}$

Step 5

The final solution is all the values that make $(\ln (x) - 2) (\ln (x) + 1) = 0$ true.

$x = e^{2}, \frac{1}{e}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = e^{2}, \frac{1}{e}$

Decimal Form:

$x = 7.38905609 \dots, 0.36787944 \dots$