The derivative of a logarithmic function represents the rate of change of the function. The graph shows that the rate of change decreases as $x$ increases.

The graph of $y = ln(x)$ is the reflection of the graph of $y = e^x$ over the line $y = x$, as they are inverse functions.

Logarithmic functions are used to model situations with proportional growth or repeated multiplication, especially when dealing with large ranges of values. They help in scaling down data to a manageable level.

Sound levels are measured in decibels, which have a logarithmic relationship with sound intensity. Logarithmic functions can effectively represent the wide range of sound intensities that humans can perceive.

Logarithmic and exponential functions are inverses of each other. A logarithm asks, 'How many times do I multiply this base by itself to get this number?'

Logarithmic regression can be used to find the line of best fit for data that exhibits a logarithmic relationship. This allows for the creation of a logarithmic function model that can predict values.

The reference intensity ($I_0$) represents the threshold of human hearing. It provides a baseline for comparing different sound intensities and calculating sound levels in decibels.

It means that the light intensity diminishes more rapidly at shallower depths, and the rate of decrease slows down as you go deeper. This is because the initial layers of water absorb more light.

Logarithmic functions are used because they can model phenomena where changes in one variable result in proportional changes in another, especially when dealing with large ranges of values. This is common in many real-world scenarios.

A function that is the inverse of an exponential function, used to model proportional growth or repeated multiplication.

A unit used to measure sound level, which has a logarithmic relationship with sound intensity.

The power of sound per unit area, typically measured in watts per square meter ($W/m^2$).

Reference intensity, the threshold of human hearing, approximately $10^{-12} W/m^2$.

A statistical method used to model the relationship between variables when one variable changes logarithmically with respect to the other.

A statistical measure that represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model.