Question about logarithmic differentiation

[Mr Davis 97]

I have a question about logarithmic differentiation, especially concerning the absolute value involved. For example, if we have the function $y = 2^x \sin x$, the domain is all real numbers. So what happens when we take $\ln$ of both sides of the equation? The antilogarithm must be greater than $0$, but the domain of the function is all real numbers. How do we account for this? I know the answer has something to do with absolute value, but I am not sure how...

 

[mathman]

Ln of negative numbers gets you into complex analysis. I suggest you make your question more specific.

 

[Mark44]

Assuming that you're concerned only with real-valued functions, since the domain of ln is positive reals, $\ln(2^x\sin(x))$ is defined only for intervals in which sin(x) > 0, open intervals of the form $(2k\pi, (2k + 1)\pi)$, with k in the integers. For the other intervals, $\ln(2^x\sin(x))$ is not defined, resulting in gaps in the graph.