Natural Log : seems as a discontinous function

In summary, a function is continuous at a point c in its domain if the limit of the function as x approaches c exists and is equal to the function at that point. However, for ln(x), the natural log of x, this is not the case at x=0 since ln(0) is not defined. Therefore, ln(x) is not continuous at x=0 and its domain is limited to positive numbers. This applies to all logarithmic functions, as including 0 in the domain would make the function discontinuous.
  • #1
hms.tech
247
0
The function f is continuous at some point c of its domain if the limit of f(x) as x approaches c through the domain of f exists and is equal to f(c). In mathematical notation, this is written as
[itex]lim_{x\rightarrow c}[/itex] f(x) = f(c) from the positive and negative sides .

For ln(x) (the natural log of x), as x[itex]\rightarrow[/itex]0 , ln(x) approaches -[itex]\infty[/itex]

Hence I would stand by the notion that ln(x) is not continuous since at x=0, the function is not defined .

Also, the graph does not exist in x<0 domain ; so ln(x) can never approach -[itex]\infty[/itex] from the -ve "x" side .
 
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  • #2
A function is continuous if it continuous at every point in its domain. The domain of natural logarithm (when treated as a real function) is ##(0,\infty)##. Zero is not part of the domain of log.
 
  • #3
The part you seem to forget is "some point c of its domain". 0 is not in the domain of ln so this is why your observation is not a problem. The domain of ln is the positive numbers.

It is however an interesting observation in its own right and it implies that you cannot possibly define ln(0) in such a way that the extension would be continuous.
 
  • #4
extending the idea you proposed, would it be logical to assume that each and every log(x) function has a domain (0,infinity) ?
 
  • #5
hms.tech said:
extending the idea you proposed, would it be logical to assume that each and every log(x) function has a domain (0,infinity) ?

Yes (slight lie, see below). Every log is defined precisely on the positive numbers.
This is a slight lie because in complex analysis we actually extend log and define it for many complex numbers. However we will never be able to include 0 in the domain (without making it discontinuous), and this little side remark is not important to you right now.
 

What is a natural log?

A natural log, also known as the logarithm with base e, is a mathematical function that represents the inverse of the exponential function. It is commonly used in mathematics and science to solve for unknown exponential variables.

Why does the natural log seem to be a discontinuous function?

The natural log function is not actually discontinuous, but it may appear that way due to its shape on a graph. This is because the natural log has a vertical asymptote at x=0, which means the function cannot be evaluated at that point. However, the function is still continuous at all other points.

How is the natural log related to e?

The natural log is defined as the logarithm with base e, where e is a mathematical constant approximately equal to 2.718. It is the base that makes the derivative of the natural log function equal to 1, making it a useful tool in calculus and other mathematical applications.

What is the domain and range of the natural log function?

The domain of the natural log function is all positive real numbers, excluding zero. This is because the function cannot be evaluated at x=0. The range of the natural log function is all real numbers, as the function can output both positive and negative values.

How is the natural log used in science?

The natural log is commonly used in science to analyze exponential growth and decay. It is also used in various scientific formulas and equations, such as in physics and chemistry, to solve for unknown variables. Additionally, the natural log plays an important role in statistics, where it is used to transform data in order to make it more normally distributed.

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