# Logarithms A [geometric-progression](geometric-progression.md) can be written as: $a, ar, ar^2, ar^3, \dots$ So, for instance, if $r = \frac{1}{2}$, we have: $1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}$ And if $r =2$: $1, 2, 4, 8, 16$ This differs from an [Arithmetic Progression](Arithmetic%20Progression.md), with is a sequence of numbers with a common difference. For example: $1, 2, 3, 4, 5, 6, 7, \dots$ Or: $7, 9, 11, 13, 15, \dots$ # Note for e, the story of a number (page 121) We see that the logarithmic spiral has the equation: $ln(r) = a \theta$ Rewriting, we have: $\theta = \frac{ln(r)}{a}$ We can correctly say that $\theta$ is a function of $r$: $\theta = f(r)$ Likewise, we could write this relationship as: $r = e^{a \theta}$ And here we can say that $r$ is a function of $\theta$: $r = g(\theta)$ Where $g = f^{-1}$. The key idea here is the following: > As we increase the angle $\theta$ by equal amounts, the distance $r$ from the pole increases by equal *ratios*. In other words, $r$ increases according to a *geometric progression*. The reason for pointing this out is make sure that the following intuition exists: > When we see something is increasing according to a *geometric progression*, we know that exponentiation is at play. When we see that our input is increasing according to a geometric progression but our output is increasing according to an *arithmetic progression*, we know that a logarithm is at play. ### Logarithm vs Root Consider that there are three ways to represent the relationship between $10^3 = 1000$, namely: $10^3 = 1000$ $\sqrt[3]{1000} = 10$ $log_{10}(1000) = 3$ So, now we can think about treating $10^x$ and $x^10$ as two different functions: $f(x) = 10^x$ $g(x) = x^10$ Above, we see that our variable takes a different role in $f$ and $g$. Hence, their inverses will be different: $f^{-1}(x) = log_{10}(f(x))$ $g^{-1}(x) = \sqrt[10]{f(x)}$ The key idea to keep in mind here is that: > We have *three completely different notations* for the exact same *fact/relationship*. A great intuition to take away is the following: > What the $log$ *wants to be* is the **exponent**. So, if we see a log, we should think: $\overbrace{log_{10}(x)}^{\text{wants to be an exponent}}$ --- tags: #mathematics #logarithms #series E ,the story of a number, page 7, 67, 69, 121 created: 2020-12-05 modified: 2020-12-11 #### References * [Logarithm Fundamentals - 3b1b](https://www.youtube.com/watch?v=cEvgcoyZvB4)