Glossary
$R^2$ value
A statistical measure (coefficient of determination) that represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s) in a regression model, indicating how well the model fits the data.
Example:
If a linear model predicting house prices based on square footage has an value of 0.95, it means 95% of the variation in house prices can be explained by the square footage.
Data Modeling
The process of using mathematical functions to represent and analyze patterns within a set of data, allowing for predictions and insights.
Example:
A meteorologist might use data modeling to predict future weather patterns based on historical temperature, pressure, and humidity readings.
Decibels (dB)
A logarithmic unit used to measure sound level, relating sound intensity to a reference intensity.
Example:
A quiet library might be around 40 decibels (dB), while a jet engine at takeoff can exceed 120 dB, illustrating the vast range of sound levels on a compressed scale.
Exponential Functions
Functions where the variable appears in the exponent, representing rapid growth or decay, and serving as the inverse of logarithmic functions.
Example:
The growth of bacteria in a petri dish under ideal conditions is often described by an exponential function, showing rapid increase over time.
Log Rules
Algebraic properties that govern how logarithmic expressions can be manipulated, such as the product rule, quotient rule, and power rule.
Example:
To simplify , you would apply the product log rule to get , which simplifies further to .
Logarithmic Functions
Functions that are the inverse of exponential functions, used to model situations where input values change proportionally over equal output intervals, often dealing with large ranges of values.
Example:
When tracking the acidity of a solution, a chemist uses a logarithmic function (pH scale) to represent hydrogen ion concentration, where a small change in pH indicates a large change in acidity.
Logarithmic Model
A mathematical equation, typically of the form $y = m \log_b(x) + c$ or $y = m \ln(x) + c$, used to represent relationships in data where one variable changes logarithmically with respect to another.
Example:
Scientists might use a logarithmic model to describe how the perceived brightness of a star relates to its actual luminosity, as human perception of light is logarithmic.
Logarithmic Regression
A statistical method used to find the best-fitting logarithmic curve for a given set of data points, often performed using technology like graphing calculators.
Example:
To analyze the relationship between the number of hours studied and test scores, a student might perform logarithmic regression if the scores improve rapidly at first but then level off with more study time.
Natural Log Function ($y = \ln(x)$)
A logarithmic function with base 'e' (Euler's number), commonly used in modeling natural phenomena involving continuous growth or decay.
Example:
The time it takes for a population to reach a certain size, given continuous growth, can often be modeled using the natural log function ().
Proportional Growth
A type of growth where the rate of change is proportional to the current quantity, often modeled by exponential or logarithmic functions depending on whether the input or output is changing proportionally.
Example:
The spread of a rumor might initially show proportional growth, where the number of new people hearing it is a constant percentage of those who already know.
Reference Intensity ($I_0$)
A standard baseline sound intensity, typically the threshold of human hearing ($10^{-12} W/m^2$), used as the denominator in the decibel formula to calculate sound levels.
Example:
When calculating the decibel level of a whisper, its actual intensity is compared to the reference intensity () to determine how many orders of magnitude louder it is than the quietest audible sound.
Regression Analysis
A statistical process for estimating the relationships among variables, used to determine the form of a mathematical model (e.g., linear, exponential, logarithmic) that best fits a data set.
Example:
Economists use regression analysis to understand how changes in interest rates might affect consumer spending, identifying the best-fit curve for their economic data.
Repeated Multiplication
A process where a quantity is multiplied by a constant factor multiple times, which is the basis for exponential growth and decay, and whose inverse is described by logarithmic functions.
Example:
Calculating compound interest involves repeated multiplication of the principal by a growth factor over each compounding period.
Sound Intensity
The power of a sound wave per unit area, typically measured in watts per square meter ($W/m^2$), which has a logarithmic relationship with sound level in decibels.
Example:
If you double the distance from a speaker, the sound intensity decreases significantly, following an inverse square law, which is then perceived logarithmically by the ear.