AI Math Basics: Logarithms and Exponentials, Part 1: Exponential Functions
Good morning from the Research Department at Qualiteg Inc. Today, let's study exponential functions.
They come in handy when deriving the differentiation formula for logarithmic functions, which is why it's their turn now.
What is an exponential function?
An exponential function is a mathematical expression that represents repeated multiplication. For example, consider "multiplying 2 three times": this is \(2 \times 2 \times 2\), and the result is \(8\). Mathematically, we write this as \(2^3 = 8\). Here, the form \(2^3\) is the exponential expression, where \(2\) is the base and \(3\) is the exponent.
Exponential functions are extremely important for describing phenomena seen throughout nature, science, and technology. Examples include compound interest at a bank and bacterial growth, phenomena in which the rate of increase itself speeds up over time.
Exponential functions also form the foundation of many other concepts and formulas in mathematics. In particular, they are indispensable for deriving the differentiation formula for logarithmic functions. Differentiating a logarithmic function is the calculation that gives the slope of its graph, and understanding this differentiation formula requires the properties of exponential functions.
One important property of exponential functions is that the exponential function \(e^x\) with base \(e\), Napier's constant (approximately \(2.718\)), has a derivative equal to itself, \(e^x\).
1. Formulas for exponential functions
$$
a^{x} \times a^{y} = a^{x+y} \tag{1.1}
$$
$$
\frac 1 {a^{x}} = a^{- x} \tag{1.2}
$$
$$
\frac {a^{y}} {a^{x}} = a^{y - x} \tag{1.3}
$$
$$
(a^{x}) ^{y} = a^{xy} \tag{1.4}
$$
2. Formulas for differentiating exponential functions
When \(a \gt 0 , a \neq 1 \)
$$
(a^{x})' = a^{x} \log_e {a} \tag{1.5}
$$
In particular, when \(a = e\)
$$
(e^{x})' = e^{x} \tag{1.6}
$$
\(e\) is Napier's constant (the base of the natural logarithm)
As shown in equation \((1.6)\), the property that \(e^{x}\) remains itself when differentiated is applied in all sorts of places, so make sure you have it down.
Bonus
Finally, here is a mnemonic for remembering Napier's constant.
2.71828 = 二塁から内野にわたる (nirui kara naiya ni wataru, "from second base across the infield": a Japanese wordplay mnemonic in which the sounds of the digits 2-7-1-8-2-8 form the phrase)
What kind of situation would that be, one wonders. See you next time!
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