How do I simplify this? $\displaystyle ln(x^9 \sqrt{4-x^2})$ Thanks
You know that $\displaystyle ln(ab) = ln(a) + ln(b)$ so
$\displaystyle ln(x^9 \sqrt{4-x^2}) = ln(x^9) + ln(\sqrt{4 - x^2})$
Rewrite that last term:
$\displaystyle = ln(x^9) + ln((4 - x^2)^{1/2})$
Now recall that $\displaystyle ln(a^b) = b \cdot ln(a)$, sooooooo.....
$\displaystyle = 9 \cdot ln(x) + \frac{1}{2} \cdot ln(4 - x^2)$
Now, you could finish here, but there's one last thing we can do:
$\displaystyle 4 - x^2 = (2 + x)(2 - x)$, so looking at that last term again:
$\displaystyle = 9 \cdot ln(x) + \frac{1}{2} \cdot ln((2 + x)(2 - x))$
$\displaystyle = 9 \cdot ln(x) + \frac{1}{2} \cdot ln(2 + x) + \frac{1}{2} \cdot ln(2 - x)$
-Dan
The term "simplify" is kind of hard to define. The result of simplifying an expression depends heavily on what use you have for the form of the final answer. In this case I am assuming that we are going to want the arguments of the ln function to have the lowest degree possible.
-Dan