How to calculate the value of $10^{\log_{10}(-1)}$?

Question

How to calculate the value of $10^{\log_{10}(-1)}$? Initially I assumed $10^{\log_{10}(-1)}=-1$ due to $10^{\log_{10}(x)}=x$. However, I'm unsure if this is true for negative numbers or only valid for $x>0$.

Thank you

Answer 1

The logarithm of a negative real number can be defined as some comlpex number. However, the rules are bit more contrived, e.g. the logarithm laws like $$\log (ab) = \log a + \log b$$ to which you are familiar from real analysis now only holds modulo some constant, and there is more than one meaningful way (actually infinitely many ways) of specifying values for $\log$.

The reason is Euler's identity $$e^{2\pi i}= 1\tag{1}$$ where $e$ is Euler's Number and $i$ is the Imaginary Unit that satisfies $i^2=-1$.

Now suppose a number $z\in\Bbb C$ in the complex plane is given in polar form, i.e. by means of a distance to the origin $|z|$ and an angle $\arg z$ relative to the positive real axis. Then due to (1) we have $$z = |z|\cdot e^{i\arg z} = |z|\cdot e^{i\arg z+2\pi i k} \tag{2} $$ where you have the freedom to chose an integer $k\in\Bbb Z$. This is because an angle (here: the one relative to the positive real axis) is only determined modulo $2\pi$ (which is modulo $360^\circ$).

Equation (2) allows to define a Natural Logarithm over the non-zero complex numbers as $$ \ln z = \ln|z| + i\arg z + 2\pi i\cdot k $$ However, due to the ambiguity in $k$, this is not a function! To make it a proper function, you have to fix one specific $k$, and you have to specify how angles are defined in order to give arg a meaning, for example $\arg z\in[0,2\pi)$ or $\arg z\in[-\pi,\pi)$ or whatever. Then, a functions you get for a specific choice of $k$ is called a branch of the (complex) logarithm with the most common one usually called main branch or principle branch with the choices $k=0$ and $\arg\in(-\pi,\pi]$$. The latter condition is often referred to as "where the cut (of ln) is made" or "cutting ln along the negative real axis", see branch cut.

You can now extend this from $\ln$ to $\log_{10}$.

This means you can make sense of $\log_{10}(-1)$, and whatever branch configuration you will be using, a characteristic property will always be that $$10^{\log_{10}(-1)} = -1$$