Finite Math Examples

$5 + 2 \ln (x) = 3$

Step 1

Move all terms not containing $\ln (x)$ to the right side of the equation.

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Step 1.1

Subtract $5$ from both sides of the equation.

$2 \ln (x) = 3 - 5$

Step 1.2

Subtract $5$ from $3$ .

$2 \ln (x) = - 2$

Step 2

Divide each term in $2 \ln (x) = - 2$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 \ln (x) = - 2$ by $2$ .

$\frac{2 \ln (x)}{2} = \frac{- 2}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 \ln (x)}{2} = \frac{- 2}{2}$

Step 2.2.1.2

Divide $\ln (x)$ by $1$ .

$\ln (x) = \frac{- 2}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $- 2$ by $2$ .

$\ln (x) = - 1$

Step 3

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

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

Step 4

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 5

Solve for $x$ .

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Step 5.1

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

$x = e^{- 1}$

Step 5.2

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

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.36787944 \dots$