Solve for z 9*e^(2z)=54

$9 \cdot e^{2 z} = 54$

Divide each term in $(9 \cdot e^{2 z}) = 54$ by $9$ and simplify.

Divide each term in $(9 \cdot e^{2 z}) = 54$ by $9$ .

$\frac{(9 \cdot e^{2 z})}{9} = \frac{54}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 \cdot e^{2 z}}{9} = \frac{54}{9}$

Divide $e^{2 z}$ by $1$ .

$e^{2 z} = \frac{54}{9}$

Simplify the right side.

Divide $54$ by $9$ .

$e^{2 z} = 6$

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

$\ln (e^{2 z}) = \ln (6)$

Expand the left side.

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

$2 z \ln (e) = \ln (6)$

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

$2 z \cdot 1 = \ln (6)$

Multiply $2$ by $1$ .

$2 z = \ln (6)$

Divide each term in $2 z = \ln (6)$ by $2$ and simplify.

Divide each term in $2 z = \ln (6)$ by $2$ .

$\frac{2 z}{2} = \frac{\ln (6)}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 z}{2} = \frac{\ln (6)}{2}$

Divide $z$ by $1$ .

$z = \frac{\ln (6)}{2}$

The result can be shown in multiple forms.

Exact Form:

$z = \frac{\ln (6)}{2}$

Decimal Form:

$z = 0.89587973 \dots$