Can Logarithms Be Defined for Negative Numbers in Advanced Mathematics?

[physicsdreams]

I recently had a test (precalc) where we had to solve log(x)-log(x+4)=2 for x.

The answer comes out negative.
I understand that in precalc we are defining the logarithms for just positive numbers, but-

Is it ever justified to define a logarithm for all numbers, both negative and positive?
(higher levels of math?)

Thanks

 

[miglo]

well in calculus while going over series, my professor introduced us to euler's famous identity/formula e^(ipi)+1=0, which is a pretty cool thing, you should look it up on google if you've never seen it
anyways my professor then went on that with this formula we are able to evaluate the natural logarithm at negative values
for example e^(ipi)=-1 taking the natural log of both sides you get ipi=ln(-1)
and you can do this for other negative values as well
ln(-5)=ipi+ln5
since ln(-5)=ln(-1)+ln(5)
if you'd like to learn more, your best bet would be complex analysis i believe

 

[I like Serena]

Hi physicsdreams!

Just like the square root of a negative number is defined for complex numbers, the logarithm of negative numbers, or rather of complex numbers in general, is defined.

You can find some info and pictures here:
http://mathworld.wolfram.com/NaturalLogarithm.html
Wikipedia also has a good article, but that may be more than you're bargaining for.

However, this is rather tricky, since the logarithm for complex numbers is not a normal function any more - it is a multivalued function.
In particular this means that the rules for logarithms that you're familiar with, do not work anymore.

 

[micromass]

Duplicate of https://www.physicsforums.com/showthread.php?t=558602

 

[CellCoree]

for your question. The concept of negative logarithms can be a bit confusing, so let me try to explain it in a way that makes sense.

First, let's review what a logarithm is. A logarithm is the inverse operation of an exponential function. In other words, if we have an exponential equation like y = a^x, the logarithm of y to the base a would be written as log_a(y) = x. This means that the logarithm tells us what power we need to raise the base (a) to in order to get the given number (y).

Now, when we talk about negative logarithms, we are essentially asking the question: "What power do we need to raise the base to in order to get a negative number?" This may seem confusing because we know that raising a number to a power should always result in a positive number. However, in mathematics, we can extend the concept of logarithms to include negative numbers.

In higher levels of math, we do indeed define logarithms for all numbers, both negative and positive. This is because logarithms have many practical applications in fields such as physics, engineering, and finance, where negative quantities may arise. For example, in physics, we use negative logarithms to represent decibels, which measure sound intensity.

In your specific example of solving log(x)-log(x+4)=2 for x, the solution may come out as a negative number because the logarithm of a negative number is a complex number (i.e. it has both a real and imaginary part). This may be beyond the scope of precalculus, but in higher levels of math, we learn how to work with complex numbers and use them in various applications.

In conclusion, while it may seem strange to define logarithms for negative numbers, it is justified in higher levels of math due to their practical applications. Keep exploring and learning about logarithms – they are a powerful tool in understanding the world around us.