Calculus Examples

$e^{- 3 x} (3 - e^{2 x})$

Step 1

Set $e^{- 3 x} (3 - e^{2 x})$ equal to $0$ .

$e^{- 3 x} (3 - e^{2 x}) = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$e^{- 3 x} = 0$

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

Step 2.2

Set $e^{- 3 x}$ equal to $0$ and solve for $x$ .

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Step 2.2.1

Set $e^{- 3 x}$ equal to $0$ .

$e^{- 3 x} = 0$

Step 2.2.2

Solve $e^{- 3 x} = 0$ for $x$ .

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Step 2.2.2.1

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

$\ln (e^{- 3 x}) = \ln (0)$

Step 2.2.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 2.2.2.3

There is no solution for $e^{- 3 x} = 0$

No solution

Step 2.3

Set $3 - e^{2 x}$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $3 - e^{2 x}$ equal to $0$ .

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

Step 2.3.2

Solve $3 - e^{2 x} = 0$ for $x$ .

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Step 2.3.2.1

Subtract $3$ from both sides of the equation.

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

Step 2.3.2.2

Divide each term in $- e^{2 x} = - 3$ by $- 1$ and simplify.

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Step 2.3.2.2.1

Divide each term in $- e^{2 x} = - 3$ by $- 1$ .

$\frac{- e^{2 x}}{- 1} = \frac{- 3}{- 1}$

Step 2.3.2.2.2

Simplify the left side.

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Step 2.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{e^{2 x}}{1} = \frac{- 3}{- 1}$

Step 2.3.2.2.2.2

Divide $e^{2 x}$ by $1$ .

$e^{2 x} = \frac{- 3}{- 1}$

Step 2.3.2.2.3

Simplify the right side.

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Step 2.3.2.2.3.1

Divide $- 3$ by $- 1$ .

$e^{2 x} = 3$

Step 2.3.2.3

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

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

Step 2.3.2.4

Expand the left side.

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Step 2.3.2.4.1

Expand $\ln (e^{2 x})$ by moving $2 x$ outside the logarithm.

$2 x \ln (e) = \ln (3)$

Step 2.3.2.4.2

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

$2 x \cdot 1 = \ln (3)$

Step 2.3.2.4.3

Multiply $2$ by $1$ .

$2 x = \ln (3)$

Step 2.3.2.5

Divide each term in $2 x = \ln (3)$ by $2$ and simplify.

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Step 2.3.2.5.1

Divide each term in $2 x = \ln (3)$ by $2$ .

$\frac{2 x}{2} = \frac{\ln (3)}{2}$

Step 2.3.2.5.2

Simplify the left side.

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Step 2.3.2.5.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.5.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\ln (3)}{2}$

Step 2.3.2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (3)}{2}$

Step 2.4

The final solution is all the values that make $e^{- 3 x} (3 - e^{2 x}) = 0$ true.

$x = \frac{\ln (3)}{2}$

Step 3