Solve for x 3+4e^(x+1)=11

$3 + 4 e^{x + 1} = 11$

Step 1

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

Subtract $3$ from both sides of the equation.

$4 e^{x + 1} = 11 - 3$

Subtract $3$ from $11$ .

$4 e^{x + 1} = 8$

Step 2

Divide each term in $4 e^{x + 1} = 8$ by $4$ and simplify.

Divide each term in $4 e^{x + 1} = 8$ by $4$ .

$\frac{4 e^{x + 1}}{4} = \frac{8}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 e^{x + 1}}{4} = \frac{8}{4}$

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

$e^{x + 1} = \frac{8}{4}$

Simplify the right side.

Divide $8$ by $4$ .

$e^{x + 1} = 2$

Step 3

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

$\ln (e^{x + 1}) = \ln (2)$

Step 4

Expand the left side.

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

$(x + 1) \ln (e) = \ln (2)$

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

$(x + 1) \cdot 1 = \ln (2)$

Multiply $x + 1$ by $1$ .

$x + 1 = \ln (2)$

Step 5

Subtract $1$ from both sides of the equation.

$x = \ln (2) - 1$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \ln (2) - 1$

Decimal Form:

$x = - 0.30685281 \dots$