Questions related to real and complex logarithms.

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0
votes
2answers
185 views

Bounding the series $\sum_{m\leq x,m\neq n}\frac{1}{|\log(m/n)|}$

I am trying to reproduce the following bound: $\sum_{1\leq m\leq x, m\neq n}\frac{1}{|\log(m/n)|}=O(x\log(x))$, for $x\geq 2$ and some $n$, $1\leq n\leq x$ (the implicit constant shouldn't depend on ...
0
votes
1answer
622 views

About irrational logarithms

Could someone provide, please, a proof of the theorem below? "Being $x$ and $b$ integers greater than $1$, which can not be represented as powers of the same basis (positive integer) and integer ...
2
votes
2answers
391 views

Why does an equiangular spiral become logarithmic (intuitively)?

One of the most famous 2D-curves are logarithmic spirals (or Spira mirabilis). They can be constructed by using a machinery that ensures a constant angle between the tangent and the radial lines all ...
11
votes
3answers
623 views

What is the $x$ in $\log_b x$ called?

In $b^a = x$, $b$ is the base, a is the exponent and $x$ is the result of the operation. But in its logarithm counterpart, $\log_{b}(x) = a$, $b$ is still the base, and $a$ is now the result. What is ...
18
votes
8answers
3k views

How did the notation “ln” for “log base e” become so pervasive?

Wikipedia sez: The natural logarithm of $x$ is often written "$\ln(x)$", instead of $\log_e(x)$ especially in disciplines where it isn't written "$\log(x)$". However, some mathematicians ...
47
votes
7answers
2k views

What's so “natural” about the base of natural logarithms?

There are so many available bases. Why is the strange number $e$ preferred over all else? Of course one could integrate $\frac{1}x$ and see this. But is there more to the story?
4
votes
1answer
469 views

Exponential and log functions compose to identity

How to prove that the exponential function and the logarithm function are the inverses of each other? I want it the following way. We must use the definition as power series, and must verify that all ...