Answer:
The inverse of h(x) is \(h^{-1}(x)\,=\,\frac{log\,6x}{log\,6}\)
Step-by-step explanation:
Given: h function, h(x) = \(6^{x-1}\)
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)
\(y\,=\,6^{x-1}\)
Take log on both sides, we get
\(log\,y\,=\,log\,6^{x-1}\)
now we use rule of logarithmic function in RHS, \(log\,m^n\,=\,n\,log\,m\) , we get
\(log\,y\,=\,(x-1)\,log\,6\)
\(x-1\,=\,\frac{log\,y}{log\,6}\)
\(x\,=\,\frac{log\,y}{log\,6}+1\)
\(x\,=\,\frac{log\,y+log\,6}{log\,6}\)
Now using another rule of logarithmic function \(log\,mn\,=\,log\,m+\,log\,n\) we get
\(x\,=\,\frac{log\,6y}{log\,6}\)
Therefore, The inverse of h(x) is \(h^{-1}(x)\,=\,\frac{log\,6x}{log\,6}\)
