Answer:
The inverse of h(x) is [tex]h^{-1}(x)\,=\,\frac{log\,6x}{log\,6}[/tex]
Step-by-step explanation:
Given: h function, h(x) = [tex]6^{x-1}[/tex]
To find: Inverse of h function.
We are given h function in terms of x. So we equate this function with arbitrary element say y, then convert the given function of x in terms of y.
The function we obtained in term of y is the required inverse function of h.
Consider,
y = h(x)
[tex]y\,=\,6^{x-1}[/tex]
Take log on both sides, we get
[tex]log\,y\,=\,log\,6^{x-1}[/tex]
now we use rule of logarithmic function in RHS, [tex]log\,m^n\,=\,n\,log\,m[/tex] , we get
[tex]log\,y\,=\,(x-1)\,log\,6[/tex]
[tex]x-1\,=\,\frac{log\,y}{log\,6}[/tex]
[tex]x\,=\,\frac{log\,y}{log\,6}+1[/tex]
[tex]x\,=\,\frac{log\,y+log\,6}{log\,6}[/tex]
Now using another rule of logarithmic function [tex]log\,mn\,=\,log\,m+\,log\,n[/tex] we get
[tex]x\,=\,\frac{log\,6y}{log\,6}[/tex]
Therefore, The inverse of h(x) is [tex]h^{-1}(x)\,=\,\frac{log\,6x}{log\,6}[/tex]
