Proving vector space, associativity

[karnten07]

Homework Statement

Im doing a problem where I am trying to show that an abelian group with a scalar multiplication is a vector field. I am 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 I am not very sure of my logs and exp's laws, not sure that they are even equal. Any pointers guys?

Homework Equations

The Attempt at a Solution

 

[foxjwill]

They wouldn't be equal.

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

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