Finite Math Examples

$3 e^{- x + 3} - 6 = 18$

Step 1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1

Add $6$ to both sides of the equation.

$3 e^{- x + 3} = 18 + 6$

Step 1.2

Add $18$ and $6$ .

$3 e^{- x + 3} = 24$

Step 2

Divide each term in $3 e^{- x + 3} = 24$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 e^{- x + 3} = 24$ by $3$ .

$\frac{3 e^{- x + 3}}{3} = \frac{24}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 e^{- x + 3}}{3} = \frac{24}{3}$

Step 2.2.1.2

Divide $e^{- x + 3}$ by $1$ .

$e^{- x + 3} = \frac{24}{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $24$ by $3$ .

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

Step 3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

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

Step 4

Expand the left side.

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Step 4.1

Expand $\ln (e^{- x + 3})$ by moving $- x + 3$ outside the logarithm.

$(- x + 3) \ln (e) = \ln (8)$

Step 4.2

The natural logarithm of $e$ is $1$ .

$(- x + 3) \cdot 1 = \ln (8)$

Step 4.3

Multiply $- x + 3$ by $1$ .

$- x + 3 = \ln (8)$

Step 5

Subtract $3$ from both sides of the equation.

$- x = \ln (8) - 3$

Step 6

Divide each term in $- x = \ln (8) - 3$ by $- 1$ and simplify.

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Step 6.1

Divide each term in $- x = \ln (8) - 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{\ln (8)}{- 1} + \frac{- 3}{- 1}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{\ln (8)}{- 1} + \frac{- 3}{- 1}$

Step 6.2.2

Divide $x$ by $1$ .

$x = \frac{\ln (8)}{- 1} + \frac{- 3}{- 1}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Move the negative one from the denominator of $\frac{\ln (8)}{- 1}$ .

$x = - 1 \cdot \ln (8) + \frac{- 3}{- 1}$

Step 6.3.1.2

Rewrite $- 1 \cdot \ln (8)$ as $- \ln (8)$ .

$x = - \ln (8) + \frac{- 3}{- 1}$

Step 6.3.1.3

Divide $- 3$ by $- 1$ .

$x = - \ln (8) + 3$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \ln (8) + 3$

Decimal Form:

$x = 0.92055845 \dots$