Question

What is the inverse function of $y={{7}^{x}}$?

Answer 1

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.