Calculus Examples

$5 = \frac{5}{6} \cdot e^{7 k}$

Step 1

Rewrite the equation as $\frac{5}{6} \cdot e^{7 k} = 5$ .

$\frac{5}{6} \cdot e^{7 k} = 5$

Step 2

Multiply both sides of the equation by $\frac{6}{5}$ .

$\frac{6}{5} (\frac{5}{6} \cdot e^{7 k}) = \frac{6}{5} \cdot 5$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{6}{5} (\frac{5}{6} \cdot e^{7 k})$ .

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Step 3.1.1.1

Combine $\frac{5}{6}$ and $e^{7 k}$ .

$\frac{6}{5} \cdot \frac{5 e^{7 k}}{6} = \frac{6}{5} \cdot 5$

Step 3.1.1.2

Combine.

$\frac{6 (5 e^{7 k})}{5 \cdot 6} = \frac{6}{5} \cdot 5$

Step 3.1.1.3

Cancel the common factor of $6$ .

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Step 3.1.1.3.1

Cancel the common factor.

$\frac{6 (5 e^{7 k})}{5 \cdot 6} = \frac{6}{5} \cdot 5$

Step 3.1.1.3.2

Rewrite the expression.

$\frac{5 e^{7 k}}{5} = \frac{6}{5} \cdot 5$

Step 3.1.1.4

Cancel the common factor of $5$ .

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Step 3.1.1.4.1

Cancel the common factor.

$\frac{5 e^{7 k}}{5} = \frac{6}{5} \cdot 5$

Step 3.1.1.4.2

Divide $e^{7 k}$ by $1$ .

$e^{7 k} = \frac{6}{5} \cdot 5$

Step 3.2

Simplify the right side.

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Step 3.2.1

Cancel the common factor of $5$ .

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Step 3.2.1.1

Cancel the common factor.

$e^{7 k} = \frac{6}{5} \cdot 5$

Step 3.2.1.2

Rewrite the expression.

$e^{7 k} = 6$

Step 4

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

$\ln (e^{7 k}) = \ln (6)$

Step 5

Expand the left side.

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Step 5.1

Expand $\ln (e^{7 k})$ by moving $7 k$ outside the logarithm.

$7 k \ln (e) = \ln (6)$

Step 5.2

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

$7 k \cdot 1 = \ln (6)$

Step 5.3

Multiply $7$ by $1$ .

$7 k = \ln (6)$

Step 6

Divide each term in $7 k = \ln (6)$ by $7$ and simplify.

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Step 6.1

Divide each term in $7 k = \ln (6)$ by $7$ .

$\frac{7 k}{7} = \frac{\ln (6)}{7}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $7$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{7 k}{7} = \frac{\ln (6)}{7}$

Step 6.2.1.2

Divide $k$ by $1$ .

$k = \frac{\ln (6)}{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$k = \frac{\ln (6)}{7}$

Decimal Form:

$k = 0.25596563 \dots$