#math/functions
# Logarithm
Suppose $b\neq{}1$ and $b>0$. Then the definition of $\log _{b}x$ is the "$b\text{ exponent of }x.
quot;
$
b^{y}=x⟺log_{b}x=y⟺x^{\frac{1}{y}}=b.
$
Consider $\log _{b}xy$, the $b\text{ exponent of }xy$. Since $b^{\alpha+\beta}=b^{\alpha}b^{\beta}$, we should see that with $b^{\alpha}=x$ and $b^{\beta}=y$ then $xy=b^{\alpha+\beta}$. How do we represent the exponents of $b$, $\alpha$ and $\beta$? With $\log _{b}x \text{ and }\log _{b}y$. Thus, we have the **product rule**
$\log _{b}xy=\log _{b}x+\log _{b}y$
Now consider $\log _{b}x^{y}$ which is equivalent to $\log _{b}(\underbrace{x\cdots x}_{y})$. Applying the product rule again we get the **power rule**
$
\log _{b}x^{y}=y\log _{b}x
$
Now consider $\log _{b}\frac{x}{y}$ which is equivalent to $\log _{b}xy^{-1}$. Then applying the product and power rule we get the **quotient rule**
$
\log _{b} \frac{x}{y}=\log _{b}x-\log _{b}y
$
Now consider $\log _{b}(b^{x})$. This is the $b\text{ exponent of }b^{x}$, which is simply $x$. Thus we get the **first inverse property**
$
\log _{b}(b^{x})=x
$
Now consider $b^{\log _{b}x}$. The exponent term is the $b \text{ exponent of }x$. Since this is the power of $b$ then we get the **second inverse property**
$
b^{\log _{b}x}=x
$
Now consider $\log _{b}(\log _{b}x)$. This is the $b \text{ exponent of }\log _{b}x$. The inner term is a number, thus the outer term is "$\log \text{ of a number}quot; which itself is a number. Thus it is irreducible.
---
What's the domain of $\log _{b}x$? $(0,\infty)$
What's the range of $\log _{b}x$? $\mathbb{R}$
For $\log _{b}(\log _{b}x)$, what must the inner $\log _{b}x$ satisfy? $\log _{b}x \in (0,\infty)$.
Following up, what must $x$ be bounded by? $x>1$ for $\log _{b}x \in(0,\infty)$ to hold.
Then what's the domain of $\log _{b}\log _{b}x$? $(1,\infty)$.
What's the range of $\log _{b}\log _{b}x$? $\mathbb{R}$.
For $\log _{b}\log _{b}\log _{b}(x)$, what the second log satisfy? $\log _{b}\log _{b}(x)\in (0, \infty)$.
Then what values of $\log _{b}(x)$ are acceptable? $\log _{b}(x)>1$.
Then what values of $x$ are acceptable? $x>b$.
```tikz
\begin{document}
\begin{tikzpicture}
% axes
\draw[->, thick] (-0.7,0) -- (17.8,0) node[right] {$x$};
\draw[->, thick] (0,-3.6) -- (0,3) node[above] {$y$};
% asymptote x = 1 (for log log x)
\draw[dashed, gray] (1,-3.6) -- (1,2.6);
\node[gray] at (1.5,2.85) {\small $x=1$};
% asymptote x = e (for log log log x)
\draw[dashed, green!55!black] (2.71828,-3.6) -- (2.71828,1.4);
\node[green!55!black] at (3.5,1.7) {\small $x=e$};
% tower column x = e^e : curves cut at e, 1, 0
\draw[dashed, gray!60] (15.1543,-0.25) -- (15.1543,2.95);
% y = log x
\draw[blue, very thick, domain=0.12:17, samples=240] plot (\x,{ln(\x)});
\node[blue] at (12.5,2.9) {$\log x$};
% y = log log x
\draw[red, very thick, domain=1.05:17, samples=260] plot (\x,{ln(ln(\x))});
\node[red] at (11.3,1.4) {$\log\log x$};
% y = log log log x
\draw[green!55!black, very thick, domain=2.85:17, samples=320]
plot (\x,{ln(ln(ln(\x)))});
\node[green!55!black] at (14.0,0.62) {$\log\log\log x$};
% landmark points
\fill (1,0) circle (2pt) node[above left] {\small $(1,0)$};
\fill (2.71828,0) circle (2pt) node[below right] {\small $(e,0)$};
\fill (2.71828,1) circle (2pt) node[above left] {\small $(e,1)$};
% tower column dots: (e^e,e),(e^e,1),(e^e,0)
\fill (15.1543,2.71828) circle (1.6pt)node[above right] {\small $(e^e,e)$};
\fill (15.1543,1) circle (1.6pt) node[above right] {\small $(e^e,1)$};
\fill (15.1543,0) circle (2pt) node[above right] {\small $(e^e,0)$};
% ticks
\foreach \x in {2,4,6,8,10,12,14,16}
\draw (\x,0.07)--(\x,-0.07) node[below] {\small $\x$};
\foreach \y in {-3,-2,-1,1,2}
\draw (0.07,\y)--(-0.07,\y) node[left] {\small $\y$};
\end{tikzpicture}
\end{document}
```