20190929, 12:34  #12  
Jul 2014
3·149 Posts 
Quote:
\( 64^3 = e^{3log(64)} \) the realvalued logarithm is defined by an integral and what the integral is? or rather \( 64^{1/2} = e^{(1/2)log(64)} \) ? ( as you said roots). Last fiddled with by wildrabbitt on 20190929 at 12:39 

20190929, 14:22  #13  
Feb 2017
Nowhere
3·17·113 Posts 
Quote:
The (natural) log of a positive real number x is defined as which is perfectly welldefined. (Here, "t" is what is called a "dummy variable," used simply to avoid using the same symbol to denote two different things in the same statement. It doesn't matter what symbol you use, as long as it isn't already being used for something else.) So, in particular, Making obvious substitutions, you can even demonstrate the usual "laws of logarithms" directly from this definition. For example, assuming a and b are positive real numbers, In the complex plane, this comes to grief. You can of course still write but now, unlike with the positive real numbers, there are myriad paths from 1 to z, and the answer you get depends on the "path of integration." Suppose, for example, you take the path which winds counterclockwise around the unit circle centered at the origin once, and takes you back to where you started, at z = 1. You get which isn't 0, the answer you would get by integrating over a "path" consisting of the single point t = 1. You can wind around the circle any number of times, counterclockwise or clockwise, and get any integer multiple of 2*pi*i, as a value of log(1). This problem can be avoided by making a "branch cut" emanating from 0 (say a ray), and defining a logarithm in the complement of the branch cut. This is essentially declaring by fiat that Thy path of integration shall not intersect the branch cut!. However, this can result in the usual "laws of exponents" giving wrong results. A classic example of the sort of trouble that can arise is misapplying the rule, valid for positive real a and b, that Using, on the one hand, a = 1 and b = 1; and, on the other, a = 1 and b = 1. This leads to which, upon "multiplying up" gives Misapplication of the rule has resulted in equating the two equal and opposite square roots of 1. If you allow complex exponents, things become truly bizarre. For example, which gives an infinity of real values, one for each integer value of n. Last fiddled with by Dr Sardonicus on 20190929 at 14:27 Reason: xiginf topsy 

20190929, 16:23  #14  
Jul 2014
3×149 Posts 
Thanks.
Is the case of \(i^i\) ....which gives an infinity of real values, one for each integer value of n. Is the above also due to misapplying the roots rule you mentioned? Quote:
/* damn */ Last fiddled with by wildrabbitt on 20190929 at 16:26 Reason: pasted copied latex which came out as links, and forgot to say thanks 

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