Question

Remember to show work and explain. Use the math font.

[Logarithms, Algebra 2]

Find the inverse of each function:


`1. \: y = {5}^{ (x)/(3) }`

`2. \: y = { - 6}^(x)`

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Answer 1

`\large\boxed{1.\ f^(-1)(x)=3\log_5x}\\\boxed{2.\ f^(-1)(x)=\log_6(-x)}`

Explanation:

`\log_ab=c\iff c=a^b\\\\\log_aa^n=n\\---------------------\\\\1.\\y=5^{(x)/(3)}\\\\\text{Exchange x and y}\\\\5^(y)/(3)=x\\\\\text{Solve for y:}\\\\5^(y)/(3)=x\qquad\log_5\text{of both sides}\\\\\log_55^(y)/(3)=\log_5x\Rightarrow(y)/(3)=\log_5x\qquad\text{multiply both sides by 3}\\\\y=3\log_5x\\---------------`

`2.\\y=-6^x\\\\\text{Exchange x and y:}\\\\-6^y=x\\\\\text{Solve for y:}\\\\-6^y=x\qquad\text{change the signs}\\\\6^y=-x\qquad\log_6\ \text{of both sides}\\\\\log_66^y=\log_6(-x)\Rightarrow y=\log_6(-x)`