Lecture 14 - Logarithms

Logarithms

Mathematical function that transforms number S (given that S > 0) into another number, typically log₁₀S. It is essentially asking to what power (exponent) should 10 be raised to so we can get S

Use of Logarithms

They’re typically applied to exponential curves so that it can flatter the curve and allow us to see all the data points easily at any scale. It is typically applied to the y-axis

Order of Magnitude

Essentially how by how many digits is it changing? So, if a number changes from 1 to 100, it increased by two orders of magnitude.

Logarithms for Predicting Time

Given S(t) = S₀e^rt, then:

• log₁₀ S(t) = log₁₀ S₀ + r(log₁₀ e)t

• Essentially, think of it as population size = initial population + speed x time

With this formula, we can figure out how long it will take the population size to reach some sort of value by transforming it a little bit more.

• time = distance/speed

• t_end = (log₁₀S_end - intercept)/slope

When comparing the increases from one time t to another, we would look at the fold-increases rather than actual increase

• This makes it so that an increase from say 1 to 10 compared to 1,000 to 10,000 is by 10 fold (times 10)

The difference between logarithms is related to the fold-difference between those two numbers before taking their logs