Find inverse function of your=6log to base 5(3x to third power -6)

Step-by-step solution:

1. Replace y with the function's output:

y = 6 \log_5 (3x^3 - 6)

2. Switch x and y to find the inverse:

x = 6 \log_5 (3y^3 - 6)

3. Isolate the logarithmic term by dividing both sides by 6:

\frac{x}{6} = \log_5 (3y^3 - 6)

4. Rewrite the logarithmic equation in exponential form:

5^{\frac{x}{6}} = 3y^3 - 6

5. Solve for y :

3y^3 = 5^{\frac{x}{6}} + 6

6. Divide by 3:

y^3 = \frac{5^{\frac{x}{6}} + 6}{3}

7. Take the cube root of both sides to solve for y :

y = \sqrt[3]{\frac{5^{\frac{x}{6}} + 6}{3}}

8. Replace y with f^{-1}(x) :

f^{-1}(x) = \sqrt[3]{\frac{5^{\frac{x}{6}} + 6}{3}}