How to solve the Logarithmic expression $ 4\cdot 3^{2x} =\mathrm{e}^x$?

Question

I am having trouble getting the right answer to the question.Any help would be appreciated: $$ 4\cdot 3^{2x} =\mathrm{e}^x. $$

Answer 1

Taking the $\log$ of both sides, you get

$$\log(4\cdot 3^{2x})=\log(e^x),$$

and using the properties of the logarithm, you get:

$$\log(4)+2x\log(3)=x$$

and it is now a simple linear equation for you to solve.

The final result is:

$$x=\frac{\log 4}{1-2\log 3}.$$

The properties of the logarithm I used are:

• $\log(ab)=\log a+\log b$,

• $\log(a^q)=q\log a$,

• $\log(e^x)=x$.