Question

What is the inverse function of \[y={{7}^{x}}\]?

Answer 1

Hint: inverse function is a function that reverses another function. Suppose if the function \[f\] is applied to an input \[x\] gives a result of \[y\], then applying the inverse function \[g\] to \[y\] gives the result \[x\] i.e. \[g\left( y \right)=x\] if and if only if \[f\left( x \right)=y\]. Inverse functions can be solved by replacing the variables.
For example, replace all \[x\] with \[y\] and all \[y\] with \[x\]

Complete step-by-step answer:
Now, let us find out the inverse function of \[y={{7}^{x}}\]
After replacing, we have to solve for \[y\].
We get,
\[x={{7}^{y}}\]
Now, upon using logarithmic function-
Apply log on both sides
\[\log x=\log \left( {{7}^{y}} \right)\]
We can find that \[\log \left( {{7}^{y}} \right)\] is in the form of \[\log {{a}^{b}}\]
The general formula would be \[\log {{a}^{b}}=b\log a\]
Now let us solve this according to the rule mentioned.
Then we get,
\[\log x=y\log 7\]
We can find that we have the \[\log \] function on both LHS and RHS of the equation. So we will be transposing the function from RHS to LHS.
That gives us, \[y=\dfrac{\log x}{\log 7}\]
From the given question, we can find that \[y={{7}^{x}}\].
\[\therefore \] The inverse function of \[y={{7}^{x}}\] is \[\dfrac{\log x}{\log 7}\].

Note: To find inverse of a function, it must satisfy a condition i.e. for a function \[f:X\to Y\] to have a inverse, it must have a property that for every \[Y\] in \[y\], there is exactly one \[x\] in \[X\] such that \[f\left( x \right)=y\]. This property ensures that a function \[g:Y\to X\] exists with the necessary relationship with \[f\].Consider a function\[f\], if the graph of \[f\] intersects at a horizontal line more than once, then we understand that there exists no inverse function to that function.