If we have a geometric series $(x_1, x_2, …, x_{n-1}, x_{n})$ of reason $q$, we can determine the general term formula to be:
$x_{1}q^{n-1} = x_{n}$
But by taking the logarithm of the equation we get:
$\log_{k}{x_{1}}+(n-1)\log_{k}{q} = \log_{k}{x_{n}}$
If we assume $x_1 > 0 < q$ (since we don't want to mess with complex numbers) and $k > 0$ (and different from 1 but I haven't figured out to make the inequal sign in MathJax yet)
Which is the formula of an arithmetic series of starting term $y_{1} = \log_{k}{x_{1}}$ and reason $r = \log_{k}{q}$
So my question is, would this mean that all properties that an arithmetic series has applies to the logarithm of the geometric series? Would $x_{c}x_{n-c}$ be a constant value? Would the product of the terms in the geometric series be a constant to the "sum of the arithmetic series"th power?
Best Answer
The answer to both your questions is yes (as long as you are careful to avoid messy branch points by always working with positive terms in the geometric series).
And you get the sign $\neq$ by the characters \neq