## geerky42 4 years ago $$\large b^x = b^y \Rightarrow x = y$$ iff $$\large b > 0, b \neq 1$$ Why b>0? Why not b ≠ 0?

1. freckles

(-1)^3=(-1)^5 => 3=5. True or false?

2. geerky42

Ok, how about b≠-1, 0, 1? I just don't understand why b couldn't be smaller than zero...

3. freckles

-1 is smaller than 0

4. freckles

The statement did not work for b<0

5. geerky42

except -1... Why couldn't it be smaller than zero?

6. geerky42

except -1...

7. geerky42

Why not $$b \in \mathbb{R}, b \neq -1, 0, 1$$?

8. freckles

Is the the thingy suppose to go both ways?

9. freckles

Or do you mean it just in that one way?

10. geerky42

Well, both way, I guess.

11. geerky42

I found this statement in a iPhone app called Math Formulas, I think this is wrong, but I'm not sure...

12. freckles

Well it is probably leading up to logarithms... Of course 1^n=1^m but this does not imply n=m.

13. freckles

Oh you understand why b cannot be -1,0, or 1.

14. freckles

15. freckles

$\log_b(x)=\frac{\ln(x)}{\ln(b)} , x>0, b>0, b \neq 1$

16. freckles

$b^x=b^y$ $\log_b(b^x)=\log_b(b^y)$ $x \log_b(b)=y \log_b(b)$ $x(1)=y(1)$ $x=y$

17. freckles

That is assuming b>0 and b does not equal 1.

18. geerky42

Well, this makes sense. Thanks.