Thread: laws of logarithms

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

Originally Posted by Yogi_Bear_79

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

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

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

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

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

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

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

Hello, Yogi_Bear_79!

Do you know the Laws of Logarithms?

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

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

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


Now differentiate it . . .

Originally Posted by Soroban

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

Wait, so $\displaystyle \log\frac{a}{b}=\log a-\log b$? That's useful...

Originally Posted by Quick

Wait, so $\displaystyle \log\frac{a}{b}=\log a-\log b$? That's useful...

Yes it's true and yes it's useful!

-Dan

Originally Posted by Quick

...That's useful...

Hi,

I've attached a pdf-file with the logarithm rules and the corresponding power rules. Maybe this is useful for you too.

EB