Finite Math Examples

${(\frac{1}{e})}^{- x} = {(\frac{1}{e^{9}})}^{x + 3}$

Step 1

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

$\ln ({(\frac{1}{e})}^{- x}) = \ln ({(\frac{1}{e^{9}})}^{x + 3})$

Step 2

Expand the left side.

Tap for more steps...

Step 2.1

Expand $\ln ({(\frac{1}{e})}^{- x})$ by moving $- x$ outside the logarithm.

$- x \ln (\frac{1}{e}) = \ln ({(\frac{1}{e^{9}})}^{x + 3})$

Step 2.2

Rewrite $\ln (\frac{1}{e})$ as $\ln (1) - \ln (e)$ .

$- x (\ln (1) - \ln (e)) = \ln ({(\frac{1}{e^{9}})}^{x + 3})$

Step 2.3

The natural logarithm of $1$ is $0$ .

$- x (0 - \ln (e)) = \ln ({(\frac{1}{e^{9}})}^{x + 3})$

Step 2.4

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

$- x (0 - 1 \cdot 1) = \ln ({(\frac{1}{e^{9}})}^{x + 3})$

Step 2.5

Multiply $- 1$ by $1$ .

$- x (0 - 1) = \ln ({(\frac{1}{e^{9}})}^{x + 3})$

Step 2.6

Subtract $1$ from $0$ .

$- x \cdot - 1 = \ln ({(\frac{1}{e^{9}})}^{x + 3})$

Step 3

Expand the right side.

Tap for more steps...

Step 3.1

Expand $\ln ({(\frac{1}{e^{9}})}^{x + 3})$ by moving $x + 3$ outside the logarithm.

$- x \cdot - 1 = (x + 3) \ln (\frac{1}{e^{9}})$

Step 3.2

Rewrite $\ln (\frac{1}{e^{9}})$ as $\ln (1) - \ln (e^{9})$ .

$- x \cdot - 1 = (x + 3) (\ln (1) - \ln (e^{9}))$

Step 3.3

Expand $\ln (e^{9})$ by moving $9$ outside the logarithm.

$- x \cdot - 1 = (x + 3) (\ln (1) - (9 \ln (e)))$

Step 3.4

The natural logarithm of $1$ is $0$ .

$- x \cdot - 1 = (x + 3) (0 - (9 \ln (e)))$

Step 3.5

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

$- x \cdot - 1 = (x + 3) (0 - (9 \cdot 1))$

Step 3.6

Multiply $9$ by $1$ .

$- x \cdot - 1 = (x + 3) (0 - 1 \cdot 9)$

Step 3.7

Multiply $- 1$ by $9$ .

$- x \cdot - 1 = (x + 3) (0 - 9)$

Step 3.8

Subtract $9$ from $0$ .

$- x \cdot - 1 = (x + 3) \cdot - 9$

Step 4

Simplify the left side.

Tap for more steps...

Step 4.1

Multiply $- x \cdot - 1$ .

Tap for more steps...

Step 4.1.1

Multiply $- 1$ by $- 1$ .

$1 x = (x + 3) \cdot - 9$

Step 4.1.2

Multiply $x$ by $1$ .

$x = (x + 3) \cdot - 9$

Step 5

Simplify the right side.

Tap for more steps...

Step 5.1

Simplify $(x + 3) \cdot - 9$ .

Tap for more steps...

Step 5.1.1

Apply the distributive property.

$x = x \cdot - 9 + 3 \cdot - 9$

Step 5.1.2

Simplify the expression.

Tap for more steps...

Step 5.1.2.1

Move $- 9$ to the left of $x$ .

$x = - 9 \cdot x + 3 \cdot - 9$

Step 5.1.2.2

Multiply $3$ by $- 9$ .

$x = - 9 x - 27$

Step 6

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 6.1

Add $9 x$ to both sides of the equation.

$x + 9 x = - 27$

Step 6.2

Add $x$ and $9 x$ .

$10 x = - 27$

Step 7

Divide each term in $10 x = - 27$ by $10$ and simplify.

Tap for more steps...

Step 7.1

Divide each term in $10 x = - 27$ by $10$ .

$\frac{10 x}{10} = \frac{- 27}{10}$

Step 7.2

Simplify the left side.

Tap for more steps...

Step 7.2.1

Cancel the common factor of $10$ .

Tap for more steps...

Step 7.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{- 27}{10}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 27}{10}$

Step 7.3

Simplify the right side.

Tap for more steps...

Step 7.3.1

Move the negative in front of the fraction.

$x = - \frac{27}{10}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{27}{10}$

Decimal Form:

$x = - 2.7$

Mixed Number Form:

$x = - 2 \frac{7}{10}$