Calculus Examples

$- \ln (12) = - (\frac{\ln (2)}{5.1}) t$

Step 1

Rewrite the equation as $- \frac{\ln (2)}{5.1} t = - \ln (12)$ .

$- \frac{\ln (2)}{5.1} t = - \ln (12)$

Step 2

Simplify the left side.

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Step 2.1

Simplify $- \frac{\ln (2)}{5.1} \cdot t$ .

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Step 2.1.1

Multiply by $1$ .

$- \frac{1 \ln (2)}{5.1} \cdot t = - \ln (12)$

Step 2.1.2

Factor $5.1$ out of $5.1$ .

$- \frac{1 \ln (2)}{5.1 (1)} \cdot t = - \ln (12)$

Step 2.1.3

Separate fractions.

$- (\frac{1}{5.1} \cdot \frac{\ln (2)}{1}) \cdot t = - \ln (12)$

Step 2.1.4

Divide $1$ by $5.1$ .

$- (0.19607843 \frac{\ln (2)}{1}) \cdot t = - \ln (12)$

Step 2.1.5

Divide $\ln (2)$ by $1$ .

$- (0.19607843 \ln (2)) \cdot t = - \ln (12)$

Step 2.1.6

Multiply $0.19607843$ by $- 1$ .

$- 0.19607843 \ln (2) t = - \ln (12)$

Step 3

Move all the terms containing a logarithm to the left side of the equation.

$- 0.19607843 \ln (2) t + \ln (12) = 0$

Step 4

Multiply $- 0.19607843$ by $\ln (2)$ .

$- 0.13591121 t + \ln (12) = 0$

Step 5

Subtract $\ln (12)$ from both sides of the equation.

$- 0.13591121 t = - \ln (12)$

Step 6

Divide each term in $- 0.13591121 t = - \ln (12)$ by $- 0.13591121$ and simplify.

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Step 6.1

Divide each term in $- 0.13591121 t = - \ln (12)$ by $- 0.13591121$ .

$\frac{- 0.13591121 t}{- 0.13591121} = \frac{- \ln (12)}{- 0.13591121}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $- 0.13591121$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{- 0.13591121 t}{- 0.13591121} = \frac{- \ln (12)}{- 0.13591121}$

Step 6.2.1.2

Divide $t$ by $1$ .

$t = \frac{- \ln (12)}{- 0.13591121}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Dividing two negative values results in a positive value.

$t = \frac{\ln (12)}{0.13591121}$

Step 6.3.2

Replace $e$ with an approximation.

$t = \frac{\log_{2.71828182} (12)}{0.13591121}$

Step 6.3.3

Log base $2.71828182$ of $12$ is approximately $2.48490664$ .

$t = \frac{2.48490664}{0.13591121}$

Step 6.3.4

Divide $2.48490664$ by $0.13591121$ .

$t = 18.28330875$