MHB Exponential Func: Solving ln6=ln2+ln3

[Perlita]

Hello everyone,
I was solving this problem:
Justify that ln6= ln2+ln3

So: exp(ln2+ln3)=exp(ln2)*exp(ln3)= 2*3= 6 = exp(ln6)
Till here, my work was okay.
What I didn't understand is : why should we say that the exponential function is strictly increasing over R before being able to simplify the equation and get: ln2+ln3=ln6 ??

Thanks

 

[I like Serena]

Hello everyone,
I was solving this problem:
Justify that ln6= ln2+ln3

So: exp(ln2+ln3)=exp(ln2)*exp(ln3)= 2*3= 6 = exp(ln6)
Till here, my work was okay.
What I didn't understand is : why should we say that the exponential function is strictly increasing over R before being able to simplify the equation and get: ln2+ln3=ln6 ??

Thanks

Hey Perlita! :)

What you need is that the function is invertible.
And a strictly increasing function is invertible.

To illustrate how it can go wrong when the function is not invertible, consider for instance the function given by $f(x)=x^2$.
We have $(-2)^2 = 4 = 2^2$.
But that does not imply that $-2 = 2$.