Precalculus Examples

$\ln (0.7) = \ln (e^{- 3 k})$

Step 1

Rewrite the equation as $\ln (e^{- 3 k}) = \ln (0.7)$ .

$\ln (e^{- 3 k}) = \ln (0.7)$

Step 2

Expand $\ln (e^{- 3 k})$ .

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Step 2.1

Expand $\ln (e^{- 3 k})$ by moving $- 3 k$ outside the logarithm.

$- 3 k \ln (e)$

Step 2.2

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

$- 3 k \cdot 1$

Step 2.3

Multiply $- 3$ by $1$ .

$- 3 k$

Step 3

The expanded equation is $- 3 k = \ln (0.7)$ .

$- 3 k = \ln (0.7)$

Step 4

Divide each term in $- 3 k = \ln (0.7)$ by $- 3$ and simplify.

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Step 4.1

Divide each term in $- 3 k = \ln (0.7)$ by $- 3$ .

$\frac{- 3 k}{- 3} = \frac{\ln (0.7)}{- 3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 3$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 3 k}{- 3} = \frac{\ln (0.7)}{- 3}$

Step 4.2.1.2

Divide $k$ by $1$ .

$k = \frac{\ln (0.7)}{- 3}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative in front of the fraction.

$k = - \frac{\ln (0.7)}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$k = - \frac{\ln (0.7)}{3}$

Decimal Form:

$k = 0.11889164 \dots$