Calculus Examples

$3 e^{2 t - 1} = 6$

Step 1

Divide each term in $3 e^{2 t - 1} = 6$ by $3$ and simplify.

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Step 1.1

Divide each term in $3 e^{2 t - 1} = 6$ by $3$ .

$\frac{3 e^{2 t - 1}}{3} = \frac{6}{3}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $3$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{3 e^{2 t - 1}}{3} = \frac{6}{3}$

Step 1.2.1.2

Divide $e^{2 t - 1}$ by $1$ .

$e^{2 t - 1} = \frac{6}{3}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $6$ by $3$ .

$e^{2 t - 1} = 2$

Step 2

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

$\ln (e^{2 t - 1}) = \ln (2)$

Step 3

Expand the left side.

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Step 3.1

Expand $\ln (e^{2 t - 1})$ by moving $2 t - 1$ outside the logarithm.

$(2 t - 1) \ln (e) = \ln (2)$

Step 3.2

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

$(2 t - 1) \cdot 1 = \ln (2)$

Step 3.3

Multiply $2 t - 1$ by $1$ .

$2 t - 1 = \ln (2)$

Step 4

Add $1$ to both sides of the equation.

$2 t = \ln (2) + 1$

Step 5

Divide each term in $2 t = \ln (2) + 1$ by $2$ and simplify.

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Step 5.1

Divide each term in $2 t = \ln (2) + 1$ by $2$ .

$\frac{2 t}{2} = \frac{\ln (2)}{2} + \frac{1}{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{2 t}{2} = \frac{\ln (2)}{2} + \frac{1}{2}$

Step 5.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (2)}{2} + \frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$t = \frac{\ln (2)}{2} + \frac{1}{2}$

Decimal Form:

$t = 0.84657359 \dots$