Calculus Examples

$e^{3 t} = 1000$

Step 1

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

$\ln (e^{3 t}) = \ln (1000)$

Step 2

Expand the left side.

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Step 2.1

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

$3 t \ln (e) = \ln (1000)$

Step 2.2

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

$3 t \cdot 1 = \ln (1000)$

Step 2.3

Multiply $3$ by $1$ .

$3 t = \ln (1000)$

Step 3

Divide each term in $3 t = \ln (1000)$ by $3$ and simplify.

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Step 3.1

Divide each term in $3 t = \ln (1000)$ by $3$ .

$\frac{3 t}{3} = \frac{\ln (1000)}{3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $3$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{3 t}{3} = \frac{\ln (1000)}{3}$

Step 3.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (1000)}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$t = \frac{\ln (1000)}{3}$

Decimal Form:

$t = 2.30258509 \dots$