DITF - Bases Other Than e and Applications Lesson

Bases Other Than e and Applications

Exponential Function to Base a

In addition to the definition of an exponential function as a function of the form f(x) = a^x, where a is a positive constant, a new definition is now possible based on e and the natural logarithm. If a is a positive real number (a ≠ 1) and x is any real number, then the exponential function to base a is denoted by a^x and defined by $a^x=e^{x\:\ln a}$ . If a = 1, then y = 1 is a constant function. These exponential functions adhere to the usual laws of exponents.

1. a⁰ = 1

2. a^xa^y = a^{x + y}

3. a^x/a^y = a^{x - y}

4. $\left(a^x\right)^{^y}=a^{xy}$

Change of Base Formula for Base a Logarithmic Functions

Logarithms with any suitable base can be expressed in terms of the natural logarithm function. This is possible based on the relationship that exists between logarithmic and exponential functions. For a positive base a and real numbers x and y,$y=\log ax\Leftrightarrow x=a^y$. For any positive number a (a ≠ 1), $\log_ax=\frac{\ln\left(x\right)}{\ln\left(a\right)}$. The derivation of this formula is found below.

For a positive base a and real numbers x and y, let $y=\log_ax$ then

$a^y=x\\ \ln\left(a^y\right)=\ln x\\ y\ln a=\ln x\\ y=\frac{\ln x}{\ln a}\\ \log_ax=\frac{\ln x}{\ln a}$

for any positive number a (a ≠ 1)

Derivatives and Integrals of General Exponential and Logarithmic Functions

Based on the chain rule and the definition of the exponential function given above, rules for derivatives involving aˣ, a^u , log₂x, and log₂u, and integrals containing aˣ and a^u can be found.

 

The presentation below targets derivatives of logarithmic functions with various bases, logarithmic properties, and calculator commands for the numerical derivative.

The presentation on derivatives of exponential (aˣ) and logarithmic functions (base a) foreshadows how exponential functions are applied in real-life problems.

Applications of Exponential Functions

Many familiar applications of exponential functions arise in the context of exponential growth and decay, compounded interest, birth rates, and spread of disease. Interpreting the number e as a limit is the basis for many applications. When f(x) = ln x, then f '(x) = 1/x and f '(1) = 1. This fact can be used to express the number e as a limit. Using the definition of a derivative as a limit, then the following is true.

Exponential growth is defined as a function y = y₀ e^kx where k > 0, whereas exponential decay occurs where k < 0. The following presentations explore exponential growth and decay rates.

 

The logistic curve is frequently used to model growth in biological populations for which saturation occurs. It is defined by $y=\frac{a}{1+be^{cx}}+d,\:a>0\:and\:b>0$. The graph of a logistic function increases or decreases between two horizontal asymptotes y = d and y = a + d. The graph changes curvature once, at the point where the output value equals a/2 + d. The function is increasing when c < 0, and decreasing if c > 0. Through logistic regression via graphing tools, a model can be found to represent logistic curves based on real-world data collection.

Bases Other Than e and Applications Practice

Bases Other Than e and Applications: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of finding bases other than e and applications.

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