Answer:
\(f^{-1}(x)=\frac{\log(\frac{x}{50000})}{\log 0.8}\)
Step-by-step explanation:
Given : The function \(f(x)=50000(0.8)^x\)
To find : The inverse of the given function?
Solution :
We write the given function as,
\(y=50000(0.8)^x\)
Taking log both side,
\(\log y=\log 50000+\log (0.8)^x\)
Apply logarithmic property, \(\log a^x=x\log a\)
\(\log y=\log 50000+x\log 0.8\)
\(x=\frac{\log y-\log 50000}{\log 0.8}\)
Now, interchange the value of x and y
\(y=\frac{\log x-\log 50000}{\log 0.8}\)
\(f^{-1}(x)=\frac{\log x-\log 50000}{\log 0.8}\)
\(f^{-1}(x)=\frac{\log(\frac{x}{50000})}{\log 0.8}\)
