Modelling using Exponentials and Logs

Understanding Exponenials and Logs

An exponential function is of the form f(x) = a * b^x where a is a constant and b > 0 and b ≠ 1. These functions exhibit a rapid growth or decay behaviour.
A logarithmic function is the inverse of an exponential function, expressed as f(x) = log_bx. The number b (base of the logarithm) is always positive and different from 1.
Natural logarithms are a special type, with base e (approximately equal to 2.71828). This is denoted as f(x) = ln x.
Exponential growth and decay model many real-world scenarios such as population growth and radioactive decay. For instance, if a quantity increases by a fixed percent at regular intervals, this is represented by an exponential function.

The graph of an exponential function is always increasing when b > 1 and always decreasing when 0 < b < 1. It will always cross the y-axis at (0, a).
The base of the logarithm and the base of the exponential function are the same number.
The graph of a logarithm function demonstrates a vertical asymptote at x = 0.
Logarithmic functions increase without upper bound when b > 1 and decrease without lower bound when 0 < b < 1.
The product rule, quotient rule, and power rule are useful properties for manipulating logarithmic equations.

Applying the definition and properties of logarithms is crucial when solving logarithmic equations.
Linear equations with exponents can often be solved by first transforming the original equation into a logarithmic form.
It’s essential to watch out for extraneous solutions while solving logarithmic equations. These are solutions that are correct algebraically but not in the context of the original problem.

Exponential regression is used to model situations where growth accelerates rapidly or where quantities decay. You fit an exponential function to a set of data points.
The constants a and b in the exponential function f(x) = a * b^x are determined by data from the real-world scenario being modelled.
Accurately interpret the constants and what they represent in the context of the particular problem. For instance, in a decay problem, the base of the exponent might represent the decay factor.

Logarithmic models are useful for representing phenomena that exhibit a rapid initial increase, followed by a slower, more steady growth.
The function f(x) = a + b*ln(x - c) is a logarithmic regression model, where a, b, and c are determined by the data from the real-world scenario.
Be accurate when interpreting the constants. For example, “c” could represent an initial value when the quantity started increasing.

Many physical phenomena, such as radioactive decay, growth of populations, and the cooling of objects, are modelled by exponential or logarithmic functions.
Financial models often utilise exponential and logarithmic functions for predicting growth trends, calculating interest, and evaluating investment options.
Logarithms are extensively employed in certain scientific scales, such as the Richter scale for measuring earthquake intensity and the pH scale for acidity.
The understanding and application of these models is essential in many disciplines, including physics, chemistry, economics, and computer science.