Glossary
Base of the logarithm
The fixed number used as the reference in a logarithmic scale, commonly 10 (common logarithm) or $e$ (natural logarithm).
Example:
When analyzing sound intensity, the decibel scale uses a base of the logarithm of 10, meaning each 10 dB increase represents a tenfold increase in sound power.
Exponential data
Data that exhibits a constant multiplicative rate of change over equal intervals, often seen in processes like growth or decay.
Example:
The number of bacteria in a petri dish doubling every hour represents exponential data, as the population grows by a constant factor over equal time intervals.
Linear scale
A scale where equal distances represent equal increments in value, maintaining proportionality between points.
Example:
A standard ruler uses a linear scale, where the distance between 1 inch and 2 inches is the same as the distance between 5 inches and 6 inches.
Linearizing Exponential Data
The process of transforming exponential data, typically by taking the logarithm of the dependent variable, so that it appears as a straight line when plotted on a semi-log graph.
Example:
To analyze the growth rate of a plant that doubles its height every week, you can perform linearizing exponential data by plotting the logarithm of its height against time, making the growth pattern easier to model with a straight line.
Logarithmic scale
A scale where equal distances represent equal ratios, effectively compressing large values and expanding small ones to better visualize exponential changes.
Example:
On a Richter scale, an earthquake of magnitude 7 is ten times more intense than a magnitude 6, demonstrating a logarithmic scale where each step represents a multiplicative increase.
Semi-log plot
A type of graph that uses a logarithmic scale on one axis and a linear scale on the other, ideal for visualizing data with a wide range of values and clarifying trends.
Example:
When tracking the rapid growth of a new viral video, a semi-log plot would clearly show if the views are increasing exponentially, turning a steep curve into a straight line.
Slope (in context of semi-log plot)
For an exponential model $y = ab^x$ plotted on a semi-log graph (log y-axis), the slope of the resulting straight line is $log_n(b)$, representing the rate of growth or decay.
Example:
If a semi-log plot of a decaying radioactive substance shows a slope of -0.3, it indicates that the logarithm of the substance's amount is decreasing by 0.3 units for every unit of time, directly related to its decay rate.
Y-intercept (in context of semi-log plot)
For an exponential model $y = ab^x$ plotted on a semi-log graph (log y-axis), the y-intercept of the resulting straight line is $log_n(a)$, representing the logarithm of the initial value.
Example:
On a semi-log plot showing population growth, if the line crosses the y-axis (time=0) at a log-value of 2, the y-intercept tells us that the initial population was or 100 individuals.