1. ## zeros

How many zeros does the function y = sin(lnx) have for 0<x<1 ?

2. Originally Posted by DINOCALC09
How many zeros does the function y = sin(lnx) have for 0<x<1 ?
$\displaystyle \sin (\ln x) = 0 \implies \ln x = \pi n \mbox{ where }n\in \mathbb{Z}$. Thus, $\displaystyle x = e^{\pi n}$ if $\displaystyle n=-1,-2,-3,...$ then $\displaystyle 0 < x < 1$ and it is a zero. Thus, there are infinitely many zeros.

3. Hello,

You know that if 0<x<1, ln(x)<0

But within $\displaystyle ]- \infty;0[$ there is an infinity of solutions making sin(ln(x))=0, because of the 2pi period.

4. Originally Posted by DINOCALC09
How many zeros does the function y = sin(lnx) have for 0<x<1 ?
$\displaystyle \sin(\ln|x|)=0$

$\displaystyle \ln|x| = \sinh(0) \Rightarrow \ldots -\pi, 0, \pi \ldots$

$\displaystyle x = e^{\pi n}$ for $\displaystyle n \ge 0$ where n is an integer and $\displaystyle 0 < x < 1$

Therefore $\displaystyle \pi n < 0$, so $\displaystyle n < 0$

$\displaystyle x = e^{\pi n} \forall n < 0, n \in Z$

$\displaystyle \forall$: for all* (isn't this supposed to have a horizontal line through the middle?)

However....

5. Write "\forall" in the math tags :-)

6. Originally Posted by Moo
Write "\forall" in the math tags :-)
Thanks!