Respuesta :
Answer: The log base 6 of x, which is written as [tex]\log_{6}(x)[/tex]
Work Shown:
[tex]f(x) = 6^x\\\\y = 6^x\\\\\log(y) = \log\left(6^x\right)\\\\\log(y) = x\log\left(6\right)\\\\\log(y) \div \log\left(6\right) = x\\\\x = \log(y) \div \log\left(6\right)\\\\x = \log_{6}(y)\\\\f^{-1}(x) = \log_{6}(x)\\\\[/tex]
The log base 6 of x, which is written as [tex]$\log _{6}(x)$[/tex].
Logarithmic functions
From the definition of logarithm:
[tex]x=\log _{b} y[/tex]
[tex]y=b^{x}[/tex]
In our case, we have
[tex]y=6^{x}[/tex]
which means that b = 6
therefore the inverse function is [tex]$x=\log _{6} y$[/tex]
which can be rewritten by switching x and y:
By using logarithmic functions,
[tex]y=\log _{6} x$[/tex]
[tex]f(x)=6^{x}$[/tex]
[tex]y=6^{x}$[/tex]
Taking log on both sides, we get
[tex]$\log (y)=\log \left(6^{x}\right)$[/tex]
[tex]\log (y)=x \log (6)[/tex]
Simplifying the function as
[tex]x=\frac{\log (y) }{\log (6)}[/tex]
[tex]x=\log _{6}(y)[/tex]
[tex]f^{-1}(x)=\log _{6}(x)[/tex]
The log base 6 of x, which is written as [tex]$\log _{6}(x)$[/tex].
To learn more about logarithmic functions
https://brainly.com/question/1695836
#SPJ2
