Question

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

Answer 1

Answer:

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

Step-by-step explanation:

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

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