Transformation of exponentials

Question

Find the transformation that takes $y=3^x$ to $y=\textit{e}^x$. I have tried:

Let $y=3^x$ to $y=e^{x'}$

$$\log_{3}(y)=x\quad\text{hence}\quad\log_{3}(y)=\frac{\log_{e}(y)}{\log_{e}(3)}$$

$$x\log_{e}(3)=x'$$

Gives the transformation as dilate by $\log_e(3)$ And also this:

$$3^{\log_{3}e}=e$$

$$e^x=3^{(\log_{3}e)x}$$

And the transformation is dilate by $\frac{1}{\log_3e}$

Could I please get an explanation which is right?

Answer 1

Both of your answers are correct as I pointed out in my comment:$$\log_3 e=\frac{\log_e e}{\log_e 3}=\frac{1}{\log_e 3}$$

Answer 2

Notice that $b = \exp(\ln(b))$ for all $b > 0$. From this follows that $3^x$ can be re-written as $$ 3^x = \exp( \ln(3^x) ) = \exp( x \cdot \ln(3) ), $$ where the last equality uses the logarithmic identity $\ln(b^a) = a \cdot \ln(b)$ for all $a$ and all $b > 0$.

Answer 3

It can be re-written as follows $$y=3^x=e^{\ln(3^x)}=e^{x\ln 3}=(e^x)^{\ln 3}$$