Question

the inverse function of y=e^3x is y=1/3 lnx

Answer 1

Consider the function $y=e^{3x}$

Replace $x$ and $y$ to obtain,

$$x=e^{3y}$$

Now, take log of base $e$ both sides to isolate the exponent as,

$$ln(x)=ln(e^{3y})$$

Use the property of logarithm $ln(e^{a})=a$ to extract the right side of equation from log as,

$$ln(x)=3y$$

Divide both sides by $3$ to obtain,

$$y=\frac{1}{3}ln(x)$$

Therefore, the inverse of the function $y=e^{3x}$ is $y=\frac{1}{3}ln(x)$. Hence proved.

Also, note that the function and it's inverse is symmetric about the line $y=x$ .