Precalculus Examples

$3.4 e^{- \frac{x}{3}} = 5.6$

Step 1

Divide each term in $3.4 e^{- \frac{x}{3}} = 5.6$ by $3.4$ and simplify.

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Step 1.1

Divide each term in $3.4 e^{- \frac{x}{3}} = 5.6$ by $3.4$ .

$\frac{3.4 e^{- \frac{x}{3}}}{3.4} = \frac{5.6}{3.4}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $3.4$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{3.4 e^{- \frac{x}{3}}}{3.4} = \frac{5.6}{3.4}$

Step 1.2.1.2

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

$e^{- \frac{x}{3}} = \frac{5.6}{3.4}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $5.6$ by $3.4$ .

$e^{- \frac{x}{3}} = 1.64705882$

Step 2

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

$\ln (e^{- \frac{x}{3}}) = \ln (1.64705882)$

Step 3

Expand the left side.

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Step 3.1

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

$- \frac{x}{3} \ln (e) = \ln (1.64705882)$

Step 3.2

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

$- \frac{x}{3} \cdot 1 = \ln (1.64705882)$

Step 3.3

Multiply $- 1$ by $1$ .

$- \frac{x}{3} = \ln (1.64705882)$

Step 4

Multiply both sides of the equation by $- 3$ .

$- 3 (- \frac{x}{3}) = - 3 \ln (1.64705882)$

Step 5

Simplify the left side.

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Step 5.1

Simplify $- 3 (- \frac{x}{3})$ .

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Step 5.1.1

Cancel the common factor of $3$ .

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Step 5.1.1.1

Move the leading negative in $- \frac{x}{3}$ into the numerator.

$- 3 \frac{- x}{3} = - 3 \ln (1.64705882)$

Step 5.1.1.2

Factor $3$ out of $- 3$ .

$3 (- 1) \frac{- x}{3} = - 3 \ln (1.64705882)$

Step 5.1.1.3

Cancel the common factor.

$3 \cdot - 1 \frac{- x}{3} = - 3 \ln (1.64705882)$

Step 5.1.1.4

Rewrite the expression.

$- - x = - 3 \ln (1.64705882)$

Step 5.1.2

Multiply.

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Step 5.1.2.1

Multiply $- 1$ by $- 1$ .

$1 x = - 3 \ln (1.64705882)$

Step 5.1.2.2

Multiply $x$ by $1$ .

$x = - 3 \ln (1.64705882)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - 3 \ln (1.64705882)$

Decimal Form:

$x = - 1.49697349 \dots$