Solve for a z=x*y^a

$z = x \cdot y^{a}$

Rewrite the equation as $x \cdot y^{a} = z$ .

$x \cdot y^{a} = z$

Divide each term in $(x \cdot y^{a}) = z$ by $x$ and simplify.

Divide each term in $(x \cdot y^{a}) = z$ by $x$ .

$\frac{(x \cdot y^{a})}{x} = \frac{z}{x}$

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x \cdot y^{a}}{x} = \frac{z}{x}$

Divide $y^{a}$ by $1$ .

$y^{a} = \frac{z}{x}$

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

$\ln (y^{a}) = \ln (\frac{z}{x})$

Expand $\ln (y^{a})$ by moving $a$ outside the logarithm.

$a \ln (y) = \ln (\frac{z}{x})$

Divide each term in $a \ln (y) = \ln (\frac{z}{x})$ by $\ln (y)$ and simplify.

Divide each term in $a \ln (y) = \ln (\frac{z}{x})$ by $\ln (y)$ .

$\frac{a \ln (y)}{\ln (y)} = \frac{\ln (\frac{z}{x})}{\ln (y)}$

Simplify the left side.

Cancel the common factor of $\ln (y)$ .

Cancel the common factor.

$\frac{a \ln (y)}{\ln (y)} = \frac{\ln (\frac{z}{x})}{\ln (y)}$

Divide $a$ by $1$ .

$a = \frac{\ln (\frac{z}{x})}{\ln (y)}$