2*(2,7)^-a^2=1 what is a?

2*(2.7)^{-a^2}  =  1

 

(Same answer as guest, just a different style.)

 

Divide both sides by 2:

--->   (2.7)^{-a^2}  =  1/2

Find the log of both sides:

--->   log( (2.7)^{-a^2}  )  =  log( 0.5 )

Using the property of logs:

--->   -a² · log( 2.7 )  =  log( 0.5 ) 

Divide both sides by log ( 2.7 ):

--->   -a²  =  log( 0.5 ) / log ( 2.7 ) 

--->    -a²  =  -0.6978564745

--->     a²  =  0.6978564745

Either  a =  0.835378...   or  a  =  -0.835378...

Solve for a over the real numbers:
2 2.7^(-a^2)=1

 

2 2.7^(-a^2)=(5/27)^(a^2) 2^(a^2+1):
(5/27)^(a^2) 2^(a^2+1)=1

 

Take the natural logarithm of both sides and use the identities log(a b)=log(a)+log(b) and log(a^b)=b log(a):
log(2) (a^2+1)-log(27/5) a^2=0

 

Expand and collect in terms of a:
(log(2)-log(27/5)) a^2+log(2)=0

 

Subtract log(2) from both sides:
(log(2)-log(27/5)) a^2=-log(2)

 

Divide both sides by log(2)-log(27/5):
a^2=-(log(2))/(log(2)-log(27/5))

 

Take the square root of both sides:
Answer: | 
 | a=sqrt((log(2))/(log(27/5)-log(2)))        or           a=-sqrt((log(2))/(log(27/5)-log(2)))=+-0.835378042837.........etc.