Proving vector space, associativity

karnten07

1. The problem statement, all variables and given/known data
Im doing a problem where im trying to show that an abelian group with a scalar multiplication is a vector field. Im trying to show associativity right now and just have a question:

im trying to show that exp(b.c.lnx) = b.exp(c.lnx)

But im not very sure of my logs and exp's laws, not sure that they are even equal. Any pointers guys?


2. Relevant equations



3. The attempt at a solution

foxjwill

They wouldn't be equal.

Since [tex]e^x[/tex] and [tex]\ln x[/tex] are inverses of each other, [tex]e^{\ln x} = x[/tex]. Therefore, your expressions can be simplified to [tex]x^{bc} = bx^c[/tex] which are not equal.

Also, a simple counter-example shows the same result: Taking [tex]x=3, b=2, c=1[/tex] we have [tex]3^{1\cdot 2}=2\cdot 3^1[/tex] which is obviously not true.