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