Characteristics of the parent logarithmic function

[opus]

I just started going over logarithmic functions in my text, and I have a question on a summary it gives on the parent function $f\left(x\right)=log_{b}\left(x\right)$
In the attached image, it says that "for any real number x...we see the following characteristics of $f\left(x\right)=log_{b}\left(x\right)$

My confusion is with the "for any real number x". If we were allowed to take the logarithm of any real number x, we would be allowed to take the logarithm of negative values. And if we were allowed to take the logarithm of negative values, we could have something like $f\left(-8\right)=log_{2}\left(-8\right)$. However 2 raised to any power will not give a negative value. So can we take the logarithm of all reals, or only positive values?

 

[fresh_42]

I just started going over logarithmic functions in my text, and I have a question on a summary it gives on the parent function $f\left(x\right)=log_{b}\left(x\right)$
In the attached image, it says that "for any real number x...we see the following characteristics of $f\left(x\right)=log_{b}\left(x\right)$

My confusion is with the "for any real number x". If we were allowed to take the logarithm of any real number x, we would be allowed to take the logarithm of negative values. And if we were allowed to take the logarithm of negative values, we could have something like $f\left(-8\right)=log_{2}\left(-8\right)$. However 2 raised to any power will not give a negative value. So can we take the logarithm of all reals, or only positive values?

Only of positive real numbers, that's what is meant by domain in the text, which is $(0,\infty)$. Things change if we consider complex variables and the complex logarithm. But in $\mathbb{R}$ we only can have positive values. This is why I don't like the notation one-to-one for injectivity, because although no two numbers are hit the same value, we still don't have all values available. So it should be one-of-all-allowed-to-one, but I guess this terrible habit of calling something one-to-one only if it's injective won't get extinct.

 

[opus]

So it is an implied positive real x values? Or we just know that the domain, as you stated, is $\left(0,∞\right)$, because the range of it's exponential inverse function is $\left(0,∞\right)$?

 

[fresh_42]

So it is an implied positive real x values? Or we just know that the domain, as you stated, is $\left(0,∞\right)$, because the range of it's exponential inverse function is $\left(0,∞\right)$?

I wouldn't reason with the inverse function, as it doesn't always exist, but we can always speak of domain and range. The domain are the allowed values a variable can take for a function, here the logarithm. So for all positive values $x>0$ the function $\log_b(x)$ is defined. Now the range is simply the set of values we can reach with the function: $\operatorname{range}(f) = \{y\in \mathbb{R}\, : \,y=f(x) \text{ for some }x\text{ of the domain }\}$. In general there can be more than one value $x$ which hits $f(y)$. For the range we just need at least one. There is exactly one for injective functions and we can build an inverse on the range, plus the logarithm has this property, but we wouldn't have an inverse for $x \mapsto x^2$. However, it also has a domain ($\mathbb{R}$) and a range ($\mathbb{R}^+_0$).

 

[Mark44]

So it is an implied positive real x values?

You shouldn't have to infer it. The author of the piece in the screenshot was being sloppy by saying "for any real number x". The real-valued log function that is evidently intended is defined only on the positive real numbers. There is a complex-valued log function, but that's definitely not what the author intended or showed in the pictured graphs.