laws of logarithms

**Yogi_Bear_79** · October 28th 2006, 07:19 AM

Please help with this example to apply the laws of logarithms to simplify the function $f(x) =ln \sqrt\frac{9-x^2}{4+x^2}$ . Then find it’s derivative.

**Quick** · October 28th 2006, 07:27 AM

Originally Posted by Yogi_Bear_79

Please help with this example to apply the laws of logarithms to simplify the function $f(x) =ln \sqrt\frac{9-x^2}{4+x^2}$ . Then find it’s derivative.

I can't help you with the derivative, but here's the simply part:

you have: $\ln\sqrt{\frac{9-x^2}{4+x^2}}$

rewrite: $\ln\left(\left(\frac{9-x^2}{4+x^2}\right)^{\frac{1}{2}}\right)$

Thus: $\frac{1}{2}\ln\frac{9-x^2}{4+x^2}$

rewrite: $\frac{1}{2}\ln\frac{(3+x)(3-x)}{4+x^2}$

It is possible to go farther but I don't know how that might help...

**Soroban** · October 28th 2006, 07:35 AM

Hello, Yogi_Bear_79!

Do you know the Laws of Logarithms?

Apply the laws of logarithms to simplify the function $f(x) = \ln\sqrt\frac{9-x^2}{4+x^2}$
Then find its derivative.

We have: . $f(x) \;= \;\ln\sqrt{\frac{9-x^2}{4+x^2}} \:=\:\ln\!\left(\frac{9-x^2}{4+x^2}\right)^{\frac{1}{2}} \;=\;\frac{1}{2}\cdot\ln\!\left(\frac{9-x^2}{4+x^2}\right)$

Then: . $f(x)\;=\;\frac{1}{2}\bigg[\ln(9-x^2) - \ln(4 + x^2)\bigg]$

Now differentiate it . . .