A post on Math Overflow about the relationship between powers, roots, and logs sparked the birth of a notation to harmonize these seemingly unrelated concepts. Grant Sanderson from 3Blue1Brown further popularized and named this notation as, “The Triangle of Power”
Exponents, Logs, and Roots
There are three general ways to describe the following mathematical equality: \[ \begin{gather*} \underbrace{\text{ }2 \times 2 \times 2}_\text{3} = 8 \\ \textit{(2 multiplied by itself 3 times equals 8)} \end{gather*} \]
- 2 multiplied by itself 3 times equals what? (answer is 8)
- what multiplied by itself 3 times equals 8? (answer is 2)
- multiply 2 by itself how many times to equal 8? (answer is 3)
or in matematical notation:
- Exponents \(2^3=8\)
- Logarithms \(\sqrt[3]8=2\)
- Roots \(\log_2(8) = 3\)
It is unfortunate that all three of these relationships use different notation types to represent the same truism that \(2\times2\times2=8\). Various websites and media have explored alternative ways to represent this relationship. The one explored in this article is the Triangle of Power.
The Triangle of Power… \(_{a}\stackrel{b}\triangle _{c}\)
Take the equation, \(2\times2\times2=8\). Using new notation, I will represent this relationship in a triad as so:
\[ \begin{align*} & _{2}\stackrel{3}\triangle _{8} \end{align*} \]
This means that I can represent each individual description (exponents, logs, and roots) as so:
\[ \begin{align*} & 2^3 = 8 & \Leftrightarrow {} \stackrel{3}{_2\triangle_{\phantom{c}}} = 8 \\ & \sqrt[3]{8} = 2 & \Leftrightarrow {} \stackrel{3}{_\phantom{a}\triangle_{8}} = 2\\ & \log_2\left(8\right) = 3 & \Leftrightarrow {} \stackrel{}{_2\triangle_{8}} = 3\\ \end{align*} \]
We can use this new notation with the variables \(a\), \(b\), and \(c\) to make this relationship more general. \[ \begin{align*} & a^b = c & \Leftrightarrow {} \stackrel{b}{_a\triangle_{\phantom{c}}} = c \\ & \sqrt[b]{c} = a & \Leftrightarrow {} \stackrel{b}{_\phantom{a}\triangle_{c}} = a\\ & \log_a\left(c\right) = b & \Leftrightarrow {} \stackrel{}{_a\triangle_{c}} = b \end{align*} \]
Properties of these operators (in standard notation)
\(a\) remains constant
\[ \begin{align*} &\text{Properties of Logarithms} &\text{Properties of Exponents} \\\\ & \log_a(x\times y) = \log_a(x) + \log_b(x) & a^{x + y} = a^x \times a ^y \\\\ & \log_a(\frac{x}{y}) = \log_a(x) - \log_b(x) & a^{x - y} = \frac{a^x}{a^y} \end{align*} \]
\(b\) remains constant
\[ \begin{align*} &\text{Properties of Exponents} &\text{Properties of Roots} \\\\ & xy^b = x^b \times y^b & \sqrt[b]{x\times y} = \sqrt[b]{x} \times \sqrt[b]{y} \\\\ & \left(\frac{x}{y}\right)^b = \frac{x^b}{y^b} & \sqrt[b]{\frac{x}{y}} = \frac{\sqrt[b]{x}}{\sqrt[b]{y}} \end{align*} \]
\(c\) remains constant
\[ \begin{align*} &\text{Properties of Logarithms} &\text{Properties of Roots} \\\\ & \log_{x\times y}(c) = \left(\left(\log_xc\right)^{-1} + \left(\log_yc\right)^{-1}\right)^{-1} & \sqrt[(x^{-1}+ y^{-1})^{-1}]{c} = \sqrt[x]{c} \times \sqrt[y]{c} \\\\ & \log_{\frac{x}{y}}(c) = \left(\left(\log_xc\right)^{-1} - \left(\log_yc\right)^{-1}\right)^{-1} & \sqrt[(x^{-1}- y^{-1})^{-1}]{c} = \frac{\sqrt[x]{c}}{\sqrt[y]{c}} \end{align*} \]
Properties of these operators (in power triangle notation)
\(a\) remains constant
\[ \begin{align*} &\text{Properties of Logarithms} &\text{Properties of Exponents} \\\\ & _{a} \stackrel{\phantom{b}} \triangle _{xy} ={} _{a}\stackrel{\phantom{b}}\triangle _{x} +{} _{a}\stackrel{\phantom{b}}\triangle _{y} & _{a} \stackrel{x+y} \triangle _{\phantom{c}} ={} _{a}\stackrel{x}\triangle _{\phantom{c}} \times{} _{a}\stackrel{y}\triangle _{\phantom{c}} \\\\ &_{a} \stackrel{\phantom{b}} \triangle _{\frac{x}{y}} ={} _{a}\stackrel{\phantom{b}}\triangle _{x} -{} _{a}\stackrel{\phantom{b}}\triangle _{y} & _{a} \stackrel{x-y} \triangle _{\phantom{c}} ={} \frac {_{a}\stackrel{x}\triangle _{\phantom{c}}} {_{a}\stackrel{y}\triangle _{\phantom{c}}} \end{align*} \]
\(b\) remains constant
\[ \begin{align*} &\text{Properties of Exponents} &\text{Properties of Roots} \\\\ & _{x\times y} \stackrel{b} \triangle _{\phantom{c}} ={} _{x}\stackrel{b}\triangle _{} \times{} _{y}\stackrel{b}\triangle _{} & _{\phantom{a}} \stackrel{b} \triangle _{x\times y} ={} _{\phantom{x}}\stackrel{b}\triangle _{x} \times{} _{\phantom{x}}\stackrel{b}\triangle _{y} \\\\ &_{\frac{x}{y}} \stackrel{b} \triangle _{\phantom{c}} ={} \frac {_{x}\stackrel{b}\triangle _{\phantom{c}} } {_{y}\stackrel{b}\triangle _{\phantom{c}} } & _{\phantom{a}} \stackrel{b} \triangle _{\frac{x}{y}} = \frac {_{\phantom{a}}\stackrel{b}\triangle _{x}} {_{\phantom{a}}\stackrel{b}\triangle _{y}} \end{align*} \]
\(c\) remains constant
\[ \begin{align*} &\text{Properties of Logarithms} &\text{Properties of Roots} \\\\ & _{x\times y} \stackrel{\phantom{b}} \triangle _{c} ={} \left( \left( _{x}\stackrel{}\triangle _{c}\right)^{-1} +{} \left(_{y}\stackrel{}\triangle _{c}\right)^{-1} \right)^{-1} & _{\phantom{a}} \stackrel{ \left(x^{-1}+y^{-1}\right)^{-1} } \triangle _{c} ={} _{\phantom{x}}\stackrel{x}\triangle _{c} \times{} _{\phantom{x}}\stackrel{y}\triangle _{c} \\\\ & _{\frac{x}{y}} \stackrel{\phantom{b}} \triangle _{c} ={} \left( \left( _{x}\stackrel{}\triangle _{c}\right)^{-1} -{} \left(_{y}\stackrel{}\triangle _{c}\right)^{-1} \right)^{-1} & _{\phantom{a}} \stackrel{ \left(x^{-1}-y^{-1}\right)^{-1} } \triangle _{c} ={} \frac {_{\phantom{x}}\stackrel{x}\triangle _{c} } {_{\phantom{x}}\stackrel{y}\triangle _{c}} \end{align*} \]
Inverse identities
Inverse identities are as follows below:
\[ \begin{align*} a &= \sqrt[b]{a^b} &=\sqrt[log_a(c)]{c} \\\\ b &= \log_a(a^b) &=\log_{\sqrt[a]{c}}(c) \\\\ c &= (\sqrt[b]{c})^c &=a^{\log_a(c)} \\\\ \end{align*} \]
Inverse identities
Inverse identities are as follows below:
e.g. \(a = f(f^{-1}(a))\)
\[ \begin{align*} a &= \text{ } _{\phantom{a}} \stackrel{b} \triangle _{ _{a}\stackrel{b}\triangle _{\phantom{c}} } \text{ } &= \text{ } _{\phantom{a}} \stackrel{_{a}\stackrel{\phantom{b}}\triangle _{c}} \triangle _{c} \\\\ b &= \text{ } _{ _{\phantom{a}}\stackrel{b}\triangle _{c} } \stackrel{\phantom{b}} \triangle _{c} \text{ } &= \text{ } _{a} \stackrel{\phantom{b}} \triangle _{ _{a}\stackrel{b}\triangle _{\phantom{c}} } \\\\ c &= \text{ } _{ _{\phantom{a}}\stackrel{b}\triangle _{c} } \stackrel{b} \triangle _{\phantom{c}} \text{ } &= \text{ } _{a} \stackrel{ _{a}\stackrel{\phantom{b}}\triangle _{c} } \triangle _{\phantom{c}} \\\\ \end{align*} \]