Precalculus Examples

$1000 e^{- 4 x} = 2400$

Step 1

Divide each term in $1000 e^{- 4 x} = 2400$ by $1000$ and simplify.

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Step 1.1

Divide each term in $1000 e^{- 4 x} = 2400$ by $1000$ .

$\frac{1000 e^{- 4 x}}{1000} = \frac{2400}{1000}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $1000$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{1000 e^{- 4 x}}{1000} = \frac{2400}{1000}$

Step 1.2.1.2

Divide $e^{- 4 x}$ by $1$ .

$e^{- 4 x} = \frac{2400}{1000}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $2400$ and $1000$ .

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Step 1.3.1.1

Factor $200$ out of $2400$ .

$e^{- 4 x} = \frac{200 (12)}{1000}$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $200$ out of $1000$ .

$e^{- 4 x} = \frac{200 \cdot 12}{200 \cdot 5}$

Step 1.3.1.2.2

Cancel the common factor.

$e^{- 4 x} = \frac{200 \cdot 12}{200 \cdot 5}$

Step 1.3.1.2.3

Rewrite the expression.

$e^{- 4 x} = \frac{12}{5}$

Step 2

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

$\ln (e^{- 4 x}) = \ln (\frac{12}{5})$

Step 3

Expand the left side.

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Step 3.1

Expand $\ln (e^{- 4 x})$ by moving $- 4 x$ outside the logarithm.

$- 4 x \ln (e) = \ln (\frac{12}{5})$

Step 3.2

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

$- 4 x \cdot 1 = \ln (\frac{12}{5})$

Step 3.3

Multiply $- 4$ by $1$ .

$- 4 x = \ln (\frac{12}{5})$

Step 4

Divide each term in $- 4 x = \ln (\frac{12}{5})$ by $- 4$ and simplify.

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Step 4.1

Divide each term in $- 4 x = \ln (\frac{12}{5})$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{\ln (\frac{12}{5})}{- 4}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 4$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{\ln (\frac{12}{5})}{- 4}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{12}{5})}{- 4}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{12}{5})}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (\frac{12}{5})}{4}$

Decimal Form:

$x = - 0.21886718 \dots$