Solve for x 4+5e^(x+2)=11

$4 + 5 e^{x + 2} = 11$

Step 1

Move all terms not containing $x$ to the right side of the equation.

Subtract $4$ from both sides of the equation.

$5 e^{x + 2} = 11 - 4$

Subtract $4$ from $11$ .

$5 e^{x + 2} = 7$

Step 2

Divide each term in $5 e^{x + 2} = 7$ by $5$ and simplify.

Divide each term in $5 e^{x + 2} = 7$ by $5$ .

$\frac{5 e^{x + 2}}{5} = \frac{7}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 e^{x + 2}}{5} = \frac{7}{5}$

Divide $e^{x + 2}$ by $1$ .

$e^{x + 2} = \frac{7}{5}$

Step 3

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

$\ln (e^{x + 2}) = \ln (\frac{7}{5})$

Step 4

Expand the left side.

Expand $\ln (e^{x + 2})$ by moving $x + 2$ outside the logarithm.

$(x + 2) \ln (e) = \ln (\frac{7}{5})$

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

$(x + 2) \cdot 1 = \ln (\frac{7}{5})$

Multiply $x + 2$ by $1$ .

$x + 2 = \ln (\frac{7}{5})$

Step 5

Subtract $2$ from both sides of the equation.

$x = \ln (\frac{7}{5}) - 2$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \ln (\frac{7}{5}) - 2$

Decimal Form:

$x = - 1.66352776 \dots$