need help with ln e^4

Question

anonymous · Answer

$$x=\ln e ^{4}$$$$e ^{x}=e ^{4}$$$$x =4$$

anonymous · Answer

how do they get from
$$x =\ln e ^{4}$$
to
$$e ^{x}=e ^{4}$$
sorry if its dumb question. 
I am aware that 
$$\ln e ^{4}=\log_{e}e ^{4} =4$$
I am just wondering about the algebra used here.

anonymous · Answer

they could have skipped the second step and immediately get x=4. but what they did was they took exponential of both sides which resulted in that second line

anonymous · Answer

what do u mean exponential of both sides

anonymous · Answer

wait.. cause x = ln x ?

anonymous · Answer

no.
$$x=\ln \exp(4)$$
$$\exp(x)=\exp \ln \exp(4)$$
so
$$\exp(x)=\exp(x)$$

anonymous · Answer

**$$\exp(x)=\exp(4)$$

anonymous · Answer

$$e ^{x}=e ^{\ln e ^{4}}$$

anonymous · Answer

?

anonymous · Answer

yea

anonymous · Answer

what is the rule that allows you to drop the ln from the right side exponent?

anonymous · Answer

exp and ln are inverse functions of each other so exp and ln cancel each other

anonymous · Answer

wooo awesome.. thats what i am missing in my head.  so u dont see ln in exponentials ;p

anonymous · Answer

thanks for sticking with me.. my brain gets so stuck on easy stuff... i just feel like i need to know every possible way