What is the log (logarithm) of zero, infinity , and 'minus' infinity ..?

Question

Can anyone tell ,,,with a bit of explaination please !!

Answer 1

Logarithm (with any base) is defined on the interval (0, infinity) only.

It is probably best to solve your question graphically - it will give you the best explanation.

There is no horizontal or slant asymptote => the log of infinity is infinity.

The log of 'minus' infinity is non-existent, even if we interpret it as a limit.

At 0, there is a vertical asymptote. The log itself is not defined at 0, but the limit of log (when x->0) is 'minus' infinity.

Hope this helps!

Answer 2

Log Infinity

Answer 3

Of course, you have to be careful what you mean when you throw around the term "infinity" in a mathematical question. If you're looking for a mathematically rigorous answer...well, you're unlikely to find it here. I suggest you ask your math teacher. However, if you're just looking for an intuitive answer:

The value of a logarithm:
y = log(x)
is the solution x to the equation:
a^y = x
For some base a (most commonly, 10, 2, or the number e). For your question, it doesn't matter which base you use, as long as a is positive.

To solve y=log(0), solve a^y=0. The solution is -infinity, since the limit as y-> -infinity of a^y equals zero.

To solve y=log(infinity), solve a^y=infinity. The solution is infinity, since the limit as y->infinity of a^y is infinity.

To solve y=log(-infinity), solve a^y=-infinity. This one is trickier. For simplicity, use the base e. The (non-rigorous) solution goes like this:
e^y=-inifinity
e^y = (-1) * infinity
e^y = e^(i*pi) * e^(infinity) <- non-complex infinity
e^y = e^ (i*pi + infinity)
y = infinity + i * pi

Again, these answers that I have given are not rigorous, but are simply intuitive answers to satisfy your curiosity.

Answer 4

Log Minus Log

Answer 5

it is undefined...
the reason lies in the definition of the logarithmic function.
The result of a log function is the power that 10 is raised to to get the # you are taking a log of. 10^0 =1, and there is no way to raise 10 to a power to get 0 or either infinity, therefore 0, infinity and minus infinity are undefined.

Answer 6

the logarithm of zero isn't well defined, but the limit is minus infinity.

any positive number to a very big negative number is practically zero.

The logarithm of infinity is infinity. (same explanation pretty much straightforward)

The ln of minus infinity is complex--the principle value is pi times i plus infinity.

Suppose the logarithm of infinity is infinity.

The ln of minus 1 is pi i. e to the pi i is negative 1.

The log of a times b is log a + log b

So the ln of negative infinity is the ln of negative 1 (pi i) plus the ln of infinity (infinity). You can convert to any other base using the crazy base theorem if you need other-than-natural logs.

Answer 7

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I don't think this is stated quite right. Let w =G(z) = log z. The domain in the z-plane needs to be defined as for z = re^(iΘ) with r>0 and Θ from some value α to α+2pi. Then the image in the w plane, where w=x+iy, is a horizontal strip with x from -inf. to +inf., and y from Θ to Θ+2pi.Why: Note w = log z = ln r + i Θ. So in the w plane, x = ln r and y = Θ . To see the mapping, fix Θ in the interval α to α+2pi (so y will = some fixed Θ). Now keeping Θ fixed, allow r to range from r>0 to infinity. As you do this, x = ln r will run from minus infinity to + infinity. The result is a horizontal line y= Θ that runs from minus to plus infinity. Now we pick and fix another Θ in the interval α to α+2pi and do the same thing. The result is a horizontal line given by y = the new Θ . As we allow Θ to range within the interval α to α+2pi we produce horizontal lines that fill up a strip of width 2pi ( namely α < y< α+2pi). Notice that the whole z-plane except the origin and the line Θ =α get mapped. The origin isn't mapped since log(0) is not defined, and the line Θ =α represents the needed "branch cut" in the domain plane to guarantee that the mapping is one to one onto the strip.

Answer 8

log (infinite)=infinite
log(0) is not definite but limlog(0)= - infinite
log(- infinite)=undefinite term in real number.

Answer 9

Ans
Log of ZERO with ANY BASE is ZERO
Proof:-
Let Log (0) to the Base a=x
Therefore a^x=0