rules of natural log

Question

a) ln(e^(12t)) = 12t

true or false

b) ln(11e^(t)) = 11t

true or false

I looked through my math textbook and my old notes but I still don't understand the rules of log and natural log. I don't even know where to begin to try and prove these answers. Please help, and explain with steps. Thank you!

 

Answer 1

a) ln(e^(12t)) = 12t
true 

b) ln(11e^(t)) = 11t
false ... it should be ln 11 + t

 

The log of a product equals the sum of the logs of the factors.

The log of an exponential equals the exponent (provided that the log base & exponentiation base are identical)

Answer 2

Ln(e^12t)=12t ln(e)

 

                =12t

 

True

 

b----

Ln(11e^(t)) = ln(11) + Ln(e^t)

 

                   =   Ln(11) +t Ln(e)

 

                  = Ln(11)  +t

False