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

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

Divide each term in $8 e^{\frac{x}{2}} = 1$ by $8$ and simplify.

Divide each term in $8 e^{\frac{x}{2}} = 1$ by $8$ .

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

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

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

Divide $e^{\frac{x}{2}}$ by $1$ .

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

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

$\ln (e^{\frac{x}{2}}) = \ln (\frac{1}{8})$

Expand the left side.

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

$\frac{x}{2} \ln (e) = \ln (\frac{1}{8})$

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

$\frac{x}{2} \cdot 1 = \ln (\frac{1}{8})$

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

$\frac{x}{2} = \ln (\frac{1}{8})$

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \ln (\frac{1}{8})$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{x}{2} = 2 \ln (\frac{1}{8})$

Rewrite the expression.

$x = 2 \ln (\frac{1}{8})$

The result can be shown in multiple forms.

Exact Form:

$x = 2 \ln (\frac{1}{8})$

Decimal Form:

$x = - 4.15888308 \dots$