No, not true. The domain of ##\log(x^2)## includes the negative reals as well as the positive reals.jedishrfu said:Here's the wikipedia discussion on logarithms with a chart of the function:
https://en.wikipedia.org/wiki/Logarithm
from it you can see that 0 is not a member and that its true for all ##x>0## ie there are no negative values for x.
Also ##log(x^2)## is equivalent to ##2*log(x)##
jedishrfu said:which gets you back to understanding the domain and range of ##log(x)##
Since ##x^2 + 9## > 0 for all real x, the domain of this function is ##\mathbb{R}##.rashida564 said:OK how about log(x2+9)
The function above doesn't have an asymptote.rashida564 said:does logarithmic always have asymptotes
rashida564 said:but at the same time i think all the logarithm function should be one to one function
Based on my other replies, what do you think?rashida564 said:and what about the range of logarithmic function can it be other than R
Mark44 said:No, not true. The domain of ##\log(x^2)## includes the negative reals as well as the positive reals.
The property ##\log(x^n) = n\log(x)## is applicable only for x > 0.
For clarification: The maximal (real) domain is ##\mathbb{R}##. You can define the function on all real numbers, but you don't have to.Mark44 said:Since ##x^2 + 9## > 0 for all real x, the domain of this function is ##\mathbb{R}##.
Yes, the domain of a logarithmic function can be the set of real numbers (R). The logarithmic function is defined for all positive real numbers, as well as 0. However, it is undefined for negative real numbers. This means that the domain of a logarithmic function is restricted to the set of positive real numbers and 0, which is equivalent to the set of real numbers (R).
The domain of a logarithmic function is restricted to positive real numbers and 0 because the logarithmic function is the inverse of the exponential function. The exponential function is defined for all real numbers, but its inverse, the logarithmic function, is only defined for positive real numbers. This is because the exponential function grows rapidly as the input approaches negative infinity, making it impossible for its inverse, the logarithmic function, to exist.
No, the domain of a logarithmic function cannot be extended to include negative real numbers. As explained in the previous question, the exponential function grows rapidly for negative inputs, making it impossible for the logarithmic function to exist as its inverse. Therefore, the domain of a logarithmic function remains restricted to positive real numbers and 0.
If you try to evaluate a logarithmic function with a negative input, you will encounter an undefined result. This is because the logarithmic function is only defined for positive real numbers and 0. Trying to evaluate it with a negative input will result in an error, as it is mathematically impossible to take the logarithm of a negative number.
Yes, the domain of a logarithmic function can be extended to include complex numbers. While the logarithmic function is not defined for negative real numbers, it can be extended to include complex numbers through the use of complex logarithms. These allow for the evaluation of logarithms of negative and complex numbers, but they are more complex and require knowledge of complex numbers and their properties.