Comment by westurner

5 years ago

As one shape of a curve; in a notebook that demonstrates multiple methods of curve fitting with and without a logarithmic transform.

Logarithm: https://simple.wikipedia.org/wiki/Logarithm ; https://en.wikipedia.org/wiki/Logarithm :

> In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x.

List of logarithmic identities: https://en.wikipedia.org/wiki/List_of_logarithmic_identities

List of integrals of logarithmic functions: https://en.wikipedia.org/wiki/List_of_integrals_of_logarithm...

As functions in a math library or a CAS that should implement the correct axioms correctly:

Sympy Docs > Functions > Contents: https://docs.sympy.org/latest/modules/functions/index.html#c...

sympy.functions.elementary.exponential. log(x, base=e) == log(x)/log(e), exp(), LambertW(), exp_polar() https://docs.sympy.org/latest/modules/functions/elementary.h...

"Exponential, Logarithmic and Trigonometric Integrals" sympy.functions.special.error_functions. Ei: exponential integral, li: logarithmic integral, Li: offset logarithmic integral https://docs.sympy.org/latest/modules/functions/special.html...

numpy.log. log() base e, log2(), log10(), log1p(x) == log(1 + x) https://numpy.org/doc/stable/reference/generated/numpy.log.h...

numpy.exp. exp(), expm1(x) == exp(x) - 1, exp2(x) == 2*x https://numpy.org/doc/stable/reference/generated/numpy.exp.h...

Khan Academy > Algebra 2 > Unit: Logarithms: https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:...

Khan Academy > Algebra (all content) > Unit: Exponential & logarithmic functions https://www.khanacademy.org/math/algebra-home/alg-exp-and-lo...

3blue1brown: "Logarithm Fundamentals | Lockdown math ep. 6", "What makes the natural log "natural"? | Lockdown math ep. 7" https://www.youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1b...

Feynmann Lectures 22-6: Algebra > Imaginary Exponents: https://www.feynmanlectures.caltech.edu/I_22.html#Ch22-S6

Power law functions: https://en.wikipedia.org/wiki/Power_law#Power-law_functions

In a two-body problem, of the 4-5 fundamental interactions: Gravity, Electroweak interaction, Strong interaction, Higgs interaction, a fifth force; which have constant exponential terms in their symbolic field descriptions? https://en.wikipedia.org/wiki/Fundamental_interaction#The_in...

Natural logs in natural systems:

Growth curve (biology) > Exponential growth: https://en.wikipedia.org/wiki/Growth_curve_(biology)#Exponen...

Basic reproduction number: https://en.wikipedia.org/wiki/Basic_reproduction_number

(... Growth hacking; awesome-grwoth-hacking: https://github.com/bekatom/awesome-growth-hacking )

Metcalf's law: https://en.wikipedia.org/wiki/Metcalfe%27s_law

Moore's law; doubling time: https://en.wikipedia.org/wiki/Moore's_law

A block reward halving is a doubling of difficulty. What block reward difficulty schedule would be a sufficient inverse of Moore's law?

A few queries:

logarithm cheatsheet https://www.google.com/search?q=logarithm+cheatsheet

logarithm on pinterest https://www.pinterest.com/search/pins/?q=logarithm

logarithm common core worksheet https://www.google.com/search?q=logarithm+common+core+worksh...

logarithm common core autograded exercise (... Khan Academy randomizes from a parametrized (?) test bank for unlimited retakes for Mastery Learning) https://www.google.com/search?q=logarithm+common+core+autogr...

2 comments

westurner

westurner 5 years ago

If only I had started my math career with a binder of notebooks or at least 3-hole-punched notes.

- [ ] Create a git repo with an environment.yml that contains e.g. `mamba install -y jupyter-book jupytext jupyter_contrib_extensions jupyterlab-git nbdime jupyter_console pandas matplotlib sympy altair requests-html`, build a container from said repo with repo2docker, and git commit and push changes made from within the JupyterLab instance that repo2docker layers on top of your reproducible software dependency requirement specification ("REES"). {bash/zsh, git, docker, repo2docker, jupyter, [MyST] markdown and $$ mathTeX $$; Google Colab, Kaggle Kernels, ml-workspace, JupyterLite}

"How I'm able to take notes in mathematics lectures using LaTeX and Vim" https://www.google.com/search?q=formula+for+entropy :

  S=k_{b}\ln\Omega

Entropy > Statistical mechanics: https://en.wikipedia.org/wiki/Entropy#Statistical_mechanics

SI unit for [ ] entropy: joules per kelvin (J*K*-1)

*****

In terms of specifying tasks for myself in order to learn {Logarithms,} I could use e.g. todo.txt markup to specify tasks with [project and concept] labels and contexts; but todo.txt doesn't support nested lists like markdown checkboxes with todo.txt markup and/or codelabels (if it's software math)

  - [ ] Read the Logarithms wikipedia page <url> and take +notes +math +logarithms @workstation
    - [o] Read
    - [x] BLD: mathrepo: generate from cookiecutter or nbdev
    - [ ] DOC: mathrepo: logarithm notes
    - [ ] DOC,ART: mathrepo: create exponential and logarithmic charts +logarithms @workstation
    - [ ] ENH,TST,DOC: mathrepo: logarithms with stdlib math, numpy, sympy (and *pytest* or at least `assert` assertion expressions)
    - [ ] ENH,TST,DOC: mathrepo: logarithms and exponents with NN libraries (and *pytest*)

Math (and logic; ultimately thermodynamics) transcend disciplines. To bikeshed - to worry about a name that can be sed-replaced later - but choose a good variable name now, Is 'mathrepo' the best scope for this project? Smaller dependency sets (i.e. simpler environment.yml) seem to result in less version conflicts. `conda env export --from-history; mamba env export --from-history; pip freeze; pipenv -h; poetry -h`

westurner 5 years ago
### LaTeX
$$ \log_{b} x = (b^? = x) $$ $$ 2^3 = 8 $$ $$ \log_{2} 8 = 3 $$ $$ \ln e = 1 $$ $$ \log_b(xy)=\log_b(x)+\log_b(y) $$ $ \begin{align} \textit{(1) } \log_b(xy) & = \log_b(x)+\log_b(y) \end{align} $
Sources: https://en.wikipedia.org/w/index.php?title=List_of_logarithm... ,
#### sympy2latex
What e.g. sympy2latex parses that LaTeX into, in terms of symbolic objects in an expression tree:
# install #!python -m pip install antlr4-python3-runtime sympy #!mamba install -y -q antlr-python-runtime sympy import sympy from sympy.parsing.latex import parse_latex def displaylatexexpr(latex): expr = parse_latex(latex) display(str(expr)) display(expr) return expr displaylatexexpr('\log_{2} 8')) # 'log(8, 2)' displaylatexexpr('\log_{2} 8 = 3')) # 'Eq(log(8, 2), 3)' displaylatexexpr('\log_b(xy) = \log_b(x)+\log_b(y)')) # 'Eq(log(x*y, b), log(x, b) + log(y, b))' displaylatexexpr('\log_{b} (xy) = \log_{b}(x)+\log_{b}(y)') # 'Eq(log(x*y, b), log(x, b) + log(y, b))' displaylatexexpr('\log_{2} (xy) = \log_{2}(x)+\log_{2}(y)') # 'Eq(log(x*y, 2), log(x, 2) + log(y, 2))'
### python standard library
https://docs.python.org/3/library/operator.html#operator.pow
https://docs.python.org/3/library/math.html#power-and-logari...
math. exp(x), expm1(), log(x, base=e), log1p(x), log2(x), log10(x), pow(x, y) : float, assert sqrt() == pow(x, 1/2)
## scipy
https://docs.scipy.org/doc/scipy/reference/generated/scipy.s... scipy.special. xlog1py()
https://docs.scipy.org/doc/scipy/reference/generated/scipy.s...
### sagemath
https://doc.sagemath.org/html/en/reference/functions/sage/fu...
### statsmodels
### TensorFlow https://www.tensorflow.org/api_docs/python/tf/math tf.math. log(), log1P(), log_sigmoid(), exp(), expm1()
https://keras.io/api/layers/activations/
SmoothReLU ("softplus") adds ln to the ReLU activation function, for example: https://en.wikipedia.org/wiki/Rectifier_(neural_networks)#So...
E.g. Softmax & LogSumExp also include natural logarithms in their definitions: https://en.wikipedia.org/wiki/Softmax_function
### PyTorch
https://pytorch.org/docs/stable/generated/torch.log.html torch. log(), log10(), log1p(), log2(), exp(), exp2(), expm1(); logaddexp() , logaddexp2(), logsumexp(), torch.special.xlog1py()
***
Regarding this learning process and these tools, Now I have a few replies to myself (!) in not-quite-markdown and with various headings: I should consolidate this information into a [MyST] markdown Jupyter Notebook and re-lead the whole thing. If this was decent markdown from the start, I'd have less markup work to do to create a ScholarlyArticle / Notebook.