Calculus Examples

$49 e^{x} - 7 e^{2 x} = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Rewrite $e^{2 x}$ as ${(e^{x})}^{2}$ .

$49 e^{x} - 7 {(e^{x})}^{2} = 0$

Step 1.2

Let $u = e^{x}$ . Substitute $u$ for all occurrences of $e^{x}$ .

$49 u - 7 u^{2} = 0$

Step 1.3

Factor $7 u$ out of $49 u - 7 u^{2}$ .

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Step 1.3.1

Factor $7 u$ out of $49 u$ .

$7 u (7) - 7 u^{2} = 0$

Step 1.3.2

Factor $7 u$ out of $- 7 u^{2}$ .

$7 u (7) + 7 u (- u) = 0$

Step 1.3.3

Factor $7 u$ out of $7 u (7) + 7 u (- u)$ .

$7 u (7 - u) = 0$

Step 1.4

Replace all occurrences of $u$ with $e^{x}$ .

$7 e^{x} (7 - e^{x}) = 0$

Step 2

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

$e^{x} = 0$

$7 - e^{x} = 0$

Step 3

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

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Step 3.1

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

$e^{x} = 0$

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

Undefined

Step 3.2.3

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

No solution

Step 4

Set $7 - e^{x}$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $7 - e^{x}$ equal to $0$ .

$7 - e^{x} = 0$

Step 4.2

Solve $7 - e^{x} = 0$ for $x$ .

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Step 4.2.1

Subtract $7$ from both sides of the equation.

$- e^{x} = - 7$

Step 4.2.2

Divide each term in $- e^{x} = - 7$ by $- 1$ and simplify.

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Step 4.2.2.1

Divide each term in $- e^{x} = - 7$ by $- 1$ .

$\frac{- e^{x}}{- 1} = \frac{- 7}{- 1}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{e^{x}}{1} = \frac{- 7}{- 1}$

Step 4.2.2.2.2

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

$e^{x} = \frac{- 7}{- 1}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Divide $- 7$ by $- 1$ .

$e^{x} = 7$

Step 4.2.3

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

$\ln (e^{x}) = \ln (7)$

Step 4.2.4

Expand the left side.

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Step 4.2.4.1

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

$x \ln (e) = \ln (7)$

Step 4.2.4.2

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

$x \cdot 1 = \ln (7)$

Step 4.2.4.3

Multiply $x$ by $1$ .

$x = \ln (7)$

Step 5

The final solution is all the values that make $7 e^{x} (7 - e^{x}) = 0$ true.

$x = \ln (7)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \ln (7)$

Decimal Form:

$x = 1.94591014 \dots$