Solve for x e^(x/2+1)+3=8

$e^{\frac{x}{2} + 1} + 3 = 8$

Step 1

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

Subtract $3$ from both sides of the equation.

$e^{\frac{x}{2} + 1} = 8 - 3$

Subtract $3$ from $8$ .

$e^{\frac{x}{2} + 1} = 5$

Step 2

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

$\ln (e^{\frac{x}{2} + 1}) = \ln (5)$

Step 3

Expand the left side.

Expand $\ln (e^{\frac{x}{2} + 1})$ by moving $\frac{x}{2} + 1$ outside the logarithm.

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

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

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

Multiply $\frac{x}{2} + 1$ by $1$ .

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

Step 4

Subtract $1$ from both sides of the equation.

$\frac{x}{2} = \ln (5) - 1$

Step 5

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (\ln (5) - 1)$

Step 6

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{x}{2} = 2 (\ln (5) - 1)$

Rewrite the expression.

$x = 2 (\ln (5) - 1)$

Simplify the right side.

Simplify $2 (\ln (5) - 1)$ .

Apply the distributive property.

$x = 2 \ln (5) + 2 \cdot - 1$

Multiply $2$ by $- 1$ .

$x = 2 \ln (5) - 2$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = 2 \ln (5) - 2$

Decimal Form:

$x = 1.21887582 \dots$