Trigonometry Examples

$e^{- c} = 8^{2 c}$

Step 1

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

$\ln (e^{- c}) = \ln (8^{2 c})$

Step 2

Expand the left side.

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Step 2.1

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

$- c \ln (e) = \ln (8^{2 c})$

Step 2.2

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

$- c \cdot 1 = \ln (8^{2 c})$

Step 2.3

Multiply $- 1$ by $1$ .

$- c = \ln (8^{2 c})$

Step 3

Expand $\ln (8^{2 c})$ by moving $2 c$ outside the logarithm.

$- c = 2 c \ln (8)$

Step 4

Subtract $2 c \ln (8)$ from both sides of the equation.

$- c - 2 c \ln (8) = 0$

Step 5

Factor the left side of the equation.

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Step 5.1

Factor $- c$ out of $- c - 2 c \ln (8)$ .

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Step 5.1.1

Factor $- c$ out of $- c$ .

$- c \cdot 1 - 2 c \ln (8) = 0$

Step 5.1.2

Factor $- c$ out of $- 2 c \ln (8)$ .

$- c \cdot 1 - c (2 \ln (8)) = 0$

Step 5.1.3

Factor $- c$ out of $- c (1) - c (2 \ln (8))$ .

$- c (1 + 2 \ln (8)) = 0$

Step 5.2

Simplify $2 \ln (8)$ by moving $2$ inside the logarithm.

$- c (1 + \ln (8^{2})) = 0$

Step 5.3

Raise $8$ to the power of $2$ .

$- c (1 + \ln (64)) = 0$

Step 6

Divide each term in $- c (1 + \ln (64)) = 0$ by $- 1 - \ln (64)$ and simplify.

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Step 6.1

Divide each term in $- c (1 + \ln (64)) = 0$ by $- 1 - \ln (64)$ .

$\frac{- c (1 + \ln (64))}{- 1 - \ln (64)} = \frac{0}{- 1 - \ln (64)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $1 + \ln (64)$ and $- 1 - \ln (64)$ .

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Step 6.2.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- c (- 1 (- 1) + \ln (64))}{- 1 - \ln (64)} = \frac{0}{- 1 - \ln (64)}$

Step 6.2.1.2

Factor $- 1$ out of $\ln (64)$ .

$\frac{- c (- 1 (- 1) - 1 (- \ln (64)))}{- 1 - \ln (64)} = \frac{0}{- 1 - \ln (64)}$

Step 6.2.1.3

Factor $- 1$ out of $- 1 (- 1) - 1 (- \ln (64))$ .

$\frac{- c (- 1 (- 1 - \ln (64)))}{- 1 - \ln (64)} = \frac{0}{- 1 - \ln (64)}$

Step 6.2.1.4

Cancel the common factor.

$\frac{- c (- 1 (- 1 - \ln (64)))}{- 1 - \ln (64)} = \frac{0}{- 1 - \ln (64)}$

Step 6.2.1.5

Divide $- c \cdot (- 1)$ by $1$ .

$- c \cdot (- 1) = \frac{0}{- 1 - \ln (64)}$

Step 6.2.2

Multiply $- c (- 1)$ .

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Step 6.2.2.1

Multiply $- 1$ by $- 1$ .

$1 c = \frac{0}{- 1 - \ln (64)}$

Step 6.2.2.2

Multiply $c$ by $1$ .

$c = \frac{0}{- 1 - \ln (64)}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $0$ by $- 1 - \ln (64)$ .

$c = 0$