1. ## Using Ln Rules

When you have the opportunity to use $\displaystyle \ln$ rules, then do you always use it? It seems to be the case in Calculus and Diff. Equations.

For instance, when getting the integrating factor (in Diff. Equations), then you often go from $\displaystyle e^{5\ln(b)} = e^{\ln (b)^{5}} = b^{5}$ etc...

In ln differentiation, if you see $\displaystyle \ln x^{2}$ then you make it $\displaystyle 2 \ln x$

2. ## Re: Using Ln Rules

Hello, Jason76!

When you have the opportunity to use $\displaystyle \ln$ rules, then do you always use it?
It seems to be the case in Calculus and Diff. Equations.
99% of the time, it is to our advantage to do so.

For instance, when getting the integrating factor (in Diff. Equations),
then you often go from $\displaystyle e^{5\ln(b)} = e^{\ln (b)^{5}} = b^{5}$ etc...
Of course!

In ln differentiation, if you see $\displaystyle \ln x^{2}$ then you make it $\displaystyle 2 \ln x$

This makes even more sense.

If we had: $\displaystyle y \:=\:\ln(x^4)$

. . pwe can use the Chain Rule: .$\displaystyle \frac{dy}{dx} \:=\:\frac{1}{x^4}(4x^3) \:=\: \frac{4}{x}$

Or apply the log rule first: .$\displaystyle y \:=\:4\ln(x)$

. . $\displaystyle \frac{dy}{dx} \:=\:4\left(\frac{1}{x}\right) \:=\:\frac{4}{x}$