Algebra Examples

$25 (1 - e^{t}) = 12$

Step 1

Divide each term in $25 (1 - e^{t}) = 12$ by $25$ and simplify.

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Step 1.1

Divide each term in $25 (1 - e^{t}) = 12$ by $25$ .

$\frac{25 (1 - e^{t})}{25} = \frac{12}{25}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $25$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{25 (1 - e^{t})}{25} = \frac{12}{25}$

Step 1.2.1.2

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

$1 - e^{t} = \frac{12}{25}$

Step 2

Move all terms not containing $t$ to the right side of the equation.

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Step 2.1

Subtract $1$ from both sides of the equation.

$- e^{t} = \frac{12}{25} - 1$

Step 2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$- e^{t} = \frac{12}{25} - 1 \cdot \frac{25}{25}$

Step 2.3

Combine $- 1$ and $\frac{25}{25}$ .

$- e^{t} = \frac{12}{25} + \frac{- 1 \cdot 25}{25}$

Step 2.4

Combine the numerators over the common denominator.

$- e^{t} = \frac{12 - 1 \cdot 25}{25}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 1$ by $25$ .

$- e^{t} = \frac{12 - 25}{25}$

Step 2.5.2

Subtract $25$ from $12$ .

$- e^{t} = \frac{- 13}{25}$

Step 2.6

Move the negative in front of the fraction.

$- e^{t} = - \frac{13}{25}$

Step 3

Divide each term in $- e^{t} = - \frac{13}{25}$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- e^{t} = - \frac{13}{25}$ by $- 1$ .

$\frac{- e^{t}}{- 1} = \frac{- \frac{13}{25}}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{e^{t}}{1} = \frac{- \frac{13}{25}}{- 1}$

Step 3.2.2

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

$e^{t} = \frac{- \frac{13}{25}}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Dividing two negative values results in a positive value.

$e^{t} = \frac{\frac{13}{25}}{1}$

Step 3.3.2

Divide $\frac{13}{25}$ by $1$ .

$e^{t} = \frac{13}{25}$

Step 4

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

$\ln (e^{t}) = \ln (\frac{13}{25})$

Step 5

Expand the left side.

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Step 5.1

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

$t \ln (e) = \ln (\frac{13}{25})$

Step 5.2

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

$t \cdot 1 = \ln (\frac{13}{25})$

Step 5.3

Multiply $t$ by $1$ .

$t = \ln (\frac{13}{25})$

Step 6

The result can be shown in multiple forms.

Exact Form:

$t = \ln (\frac{13}{25})$

Decimal Form:

$t = - 0.65392646 \dots$