Logarithmic Function: Can Domain of Logarithm be R?

[rashida564]

can domain of logarithm function be R .
i think it can and the same time it can't
it can like log(x²)
but at the same time i think all the logarithm function should be one to one function

 

[rashida564]

and what about the range of logarithmic function can it be other than R

 

[jedishrfu]

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 $2*log(x)$ which gets you back to understanding the domain and range of $log(x)$

 

[rashida564]

OK how about log(x²+9)

 

[rashida564]

does logarithmic always have asymptotes

 

[rashida564]

i know about logarithmic function but i want to increase my knowledge .

 

[Mark44]

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 $2*log(x)$

No, not true. The domain of $\log(x^2)$ includes the negative reals as well as the positive reals.
If n is an odd integer, the property $\log(x^n) = n\log(x)$ is applicable only for x > 0. If n is an even integer, then the only restriction on (real) x is that $x \ne 0$.

which gets you back to understanding the domain and range of $log(x)$

 

[Mark44]

OK how about log(x²+9)

Since $x^2 + 9$ > 0 for all real x, the domain of this function is $\mathbb{R}$.

does logarithmic always have asymptotes

The function above doesn't have an asymptote.

 

[Mark44]

but at the same time i think all the logarithm function should be one to one function

and what about the range of logarithmic function can it be other than R

Based on my other replies, what do you think?

 

[jedishrfu]

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.

Yes, you are right.

 

[mfb]

Since $x^2 + 9$ > 0 for all real x, the domain of this function is $\mathbb{R}$.

For clarification: The maximal (real) domain is $\mathbb{R}$. You can define the function on all real numbers, but you don't have to.

 

[rashida564]

so it can be ℝ