Solve for x ((y^(2x))/(y^6))^-1=(y^-3)^4

${(\frac{y^{2 x}}{y^{6}})}^{- 1} = {(y^{- 3})}^{4}$

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

$\ln ({(\frac{y^{2 x}}{y^{6}})}^{- 1}) = \ln ({(y^{- 3})}^{4})$

Expand the left side.

Expand $\ln ({(\frac{y^{2 x}}{y^{6}})}^{- 1})$ by moving $- 1$ outside the logarithm.

$- \ln (\frac{y^{2 x}}{y^{6}}) = \ln ({(y^{- 3})}^{4})$

Reduce the expression $\frac{y^{2 x}}{y^{6}}$ by cancelling the common factors.

Factor $y^{6}$ out of $y^{2 x}$ .

$- \ln (\frac{y^{6} y^{2 x - 6}}{y^{6}}) = \ln ({(y^{- 3})}^{4})$

Multiply by $1$ .

$- \ln (\frac{y^{6} y^{2 x - 6}}{y^{6} \cdot 1}) = \ln ({(y^{- 3})}^{4})$

Cancel the common factor.

$- \ln (\frac{y^{6} y^{2 x - 6}}{y^{6} \cdot 1}) = \ln ({(y^{- 3})}^{4})$

Rewrite the expression.

$- \ln (\frac{y^{2 x - 6}}{1}) = \ln ({(y^{- 3})}^{4})$

Divide $y^{2 x - 6}$ by $1$ .

$- \ln (y^{2 x - 6}) = \ln ({(y^{- 3})}^{4})$

Expand $\ln (y^{2 x - 6})$ by moving $2 x - 6$ outside the logarithm.

$- ((2 x - 6) \ln (y)) = \ln ({(y^{- 3})}^{4})$

Simplify the left side.

Simplify $- ((2 x - 6) \ln (y))$ .

Apply the distributive property.

$- (2 x \ln (y) - 6 \ln (y)) = \ln ({(y^{- 3})}^{4})$

Apply the distributive property.

$- (2 x \ln (y)) - (- 6 \ln (y)) = \ln ({(y^{- 3})}^{4})$

Multiply.

Multiply $2$ by $- 1$ .

$- 2 (x \ln (y)) - (- 6 \ln (y)) = \ln ({(y^{- 3})}^{4})$

Multiply $- 6$ by $- 1$ .

$- 2 x \ln (y) + 6 \ln (y) = \ln ({(y^{- 3})}^{4})$

Simplify the right side.

Simplify $\ln ({(y^{- 3})}^{4})$ .

Multiply the exponents in ${(y^{- 3})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 2 x \ln (y) + 6 \ln (y) = \ln (y^{- 3 \cdot 4})$

Multiply $- 3$ by $4$ .

$- 2 x \ln (y) + 6 \ln (y) = \ln (y^{- 12})$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 2 x \ln (y) + 6 \ln (y) = \ln (\frac{1}{y^{12}})$

Move all the terms containing a logarithm to the left side of the equation.

$- 2 x \ln (y) + 6 \ln (y) - \ln (\frac{1}{y^{12}}) = 0$

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

Subtract $6 \ln (y)$ from both sides of the equation.

$- 2 x \ln (y) - \ln (\frac{1}{y^{12}}) = - 6 \ln (y)$

Add $\ln (\frac{1}{y^{12}})$ to both sides of the equation.

$- 2 x \ln (y) = - 6 \ln (y) + \ln (\frac{1}{y^{12}})$

Divide each term in $- 2 x \ln (y) = - 6 \ln (y) + \ln (\frac{1}{y^{12}})$ by $- 2 \ln (y)$ and simplify.

Divide each term in $- 2 x \ln (y) = - 6 \ln (y) + \ln (\frac{1}{y^{12}})$ by $- 2 \ln (y)$ .

$\frac{- 2 x \ln (y)}{- 2 \ln (y)} = \frac{- 6 \ln (y)}{- 2 \ln (y)} + \frac{\ln (\frac{1}{y^{12}})}{- 2 \ln (y)}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x \ln (y)}{- 2 \ln (y)} = \frac{- 6 \ln (y)}{- 2 \ln (y)} + \frac{\ln (\frac{1}{y^{12}})}{- 2 \ln (y)}$

Rewrite the expression.

$\frac{x \ln (y)}{\ln (y)} = \frac{- 6 \ln (y)}{- 2 \ln (y)} + \frac{\ln (\frac{1}{y^{12}})}{- 2 \ln (y)}$

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

Cancel the common factor.

$\frac{x \ln (y)}{\ln (y)} = \frac{- 6 \ln (y)}{- 2 \ln (y)} + \frac{\ln (\frac{1}{y^{12}})}{- 2 \ln (y)}$

Divide $x$ by $1$ .

$x = \frac{- 6 \ln (y)}{- 2 \ln (y)} + \frac{\ln (\frac{1}{y^{12}})}{- 2 \ln (y)}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 6$ and $- 2$ .

Factor $- 2$ out of $- 6 \ln (y)$ .

$x = \frac{- 2 (3 \ln (y))}{- 2 \ln (y)} + \frac{\ln (\frac{1}{y^{12}})}{- 2 \ln (y)}$

Cancel the common factors.

Factor $- 2$ out of $- 2 \ln (y)$ .

$x = \frac{- 2 (3 \ln (y))}{- 2 \ln (y)} + \frac{\ln (\frac{1}{y^{12}})}{- 2 \ln (y)}$

Cancel the common factor.

$x = \frac{- 2 (3 \ln (y))}{- 2 \ln (y)} + \frac{\ln (\frac{1}{y^{12}})}{- 2 \ln (y)}$

Rewrite the expression.

$x = \frac{3 \ln (y)}{\ln (y)} + \frac{\ln (\frac{1}{y^{12}})}{- 2 \ln (y)}$

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

Cancel the common factor.

$x = \frac{3 \ln (y)}{\ln (y)} + \frac{\ln (\frac{1}{y^{12}})}{- 2 \ln (y)}$

Divide $3$ by $1$ .

$x = 3 + \frac{\ln (\frac{1}{y^{12}})}{- 2 \ln (y)}$

Move the negative in front of the fraction.

$x = 3 - \frac{\ln (\frac{1}{y^{12}})}{2 \ln (y)}$

Solve for x ((y^(2x))/(y^6))^-1=(y^-3)^4

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