Basic Math Examples

$s = 10 (1 - e^{K t})$

Step 1

Rewrite the equation as $10 (1 - e^{K t}) = s$ .

$10 (1 - e^{K t}) = s$

Step 2

Divide each term in $10 (1 - e^{K t}) = s$ by $10$ and simplify.

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Step 2.1

Divide each term in $10 (1 - e^{K t}) = s$ by $10$ .

$\frac{10 (1 - e^{K t})}{10} = \frac{s}{10}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $10$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{10 (1 - e^{K t})}{10} = \frac{s}{10}$

Step 2.2.1.2

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

$1 - e^{K t} = \frac{s}{10}$

Step 3

Subtract $1$ from both sides of the equation.

$- e^{K t} = \frac{s}{10} - 1$

Step 4

Divide each term in $- e^{K t} = \frac{s}{10} - 1$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- e^{K t} = \frac{s}{10} - 1$ by $- 1$ .

$\frac{- e^{K t}}{- 1} = \frac{\frac{s}{10}}{- 1} + \frac{- 1}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{e^{K t}}{1} = \frac{\frac{s}{10}}{- 1} + \frac{- 1}{- 1}$

Step 4.2.2

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

$e^{K t} = \frac{\frac{s}{10}}{- 1} + \frac{- 1}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Move the negative one from the denominator of $\frac{\frac{s}{10}}{- 1}$ .

$e^{K t} = - 1 \cdot \frac{s}{10} + \frac{- 1}{- 1}$

Step 4.3.1.2

Rewrite $- 1 \cdot \frac{s}{10}$ as $- \frac{s}{10}$ .

$e^{K t} = - \frac{s}{10} + \frac{- 1}{- 1}$

Step 4.3.1.3

Divide $- 1$ by $- 1$ .

$e^{K t} = - \frac{s}{10} + 1$

Step 5

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

$\ln (e^{K t}) = \ln (- \frac{s}{10} + 1)$

Step 6

Expand the left side.

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Step 6.1

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

$K t \ln (e) = \ln (- \frac{s}{10} + 1)$

Step 6.2

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

$K t \cdot 1 = \ln (- \frac{s}{10} + 1)$

Step 6.3

Multiply $K$ by $1$ .

$K t = \ln (- \frac{s}{10} + 1)$

Step 7

Divide each term in $K t = \ln (- \frac{s}{10} + 1)$ by $K$ and simplify.

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Step 7.1

Divide each term in $K t = \ln (- \frac{s}{10} + 1)$ by $K$ .

$\frac{K t}{K} = \frac{\ln (- \frac{s}{10} + 1)}{K}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $K$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{K t}{K} = \frac{\ln (- \frac{s}{10} + 1)}{K}$

Step 7.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (- \frac{s}{10} + 1)}{K}$