Precalculus Examples

$e^{- t} = 0.03$

Step 1

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

$\ln (e^{- t}) = \ln (0.03)$

Step 2

Expand the left side.

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Step 2.1

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

$- t \ln (e) = \ln (0.03)$

Step 2.2

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

$- t \cdot 1 = \ln (0.03)$

Step 2.3

Multiply $- 1$ by $1$ .

$- t = \ln (0.03)$

Step 3

Divide each term in $- t = \ln (0.03)$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- t = \ln (0.03)$ by $- 1$ .

$\frac{- t}{- 1} = \frac{\ln (0.03)}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{t}{1} = \frac{\ln (0.03)}{- 1}$

Step 3.2.2

Divide $t$ by $1$ .

$t = \frac{\ln (0.03)}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative one from the denominator of $\frac{\ln (0.03)}{- 1}$ .

$t = - 1 \cdot \ln (0.03)$

Step 3.3.2

Rewrite $- 1 \cdot \ln (0.03)$ as $- \ln (0.03)$ .

$t = - \ln (0.03)$

Step 4

The result can be shown in multiple forms.

Exact Form:

$t = - \ln (0.03)$

Decimal Form:

$t = 3.50655789 \dots$