Review of Logs

Appendix to Lab 1

What is a Log?

• Nothing more complicated than the inverse of an exponential!!!!!!!

What do Logs Do?

• logarithms turn multiplication into addition

• the log of a product is the sum of the logs of the components of the product

• log(a*b) = log(a) + log(b)

What does that mean???

• 5² = 25 • here we know that 2 (the power) is the logarithm

of 25 to base 5. • Symbolically, we write it as log5(25) = 2• 1000 = 103 can also be written as 3 = log101000.• Example 1: log10103.16 = 3.16 • Example 2: log5125 = log5(5³) = 3• log381 = ? is the same as 3? = 81• logbbx = x     for any base b

What Does This Really Mean??

• Imagine algae growing in a petri dish, starting from a single cell.

• After some time the cell will split to 2, then each cell will split again then split again, so that the total population of cells does not increase linearly (in an additive manner) through time but multiplicatively (by doubling).

• If you were to plot the number of cells through time, it would increase geometrically, not linearly.

• Logarithms are a way to rescale something which is increasing (or decreasing) in a multiplicative manner so as to make it increase (or decrease) linearly.

Why Use Logs???

• To model many natural processes, particularly in living systems.

• Our perceptions are not tuned to detect "additive differences" but rather to detect "multiplicative differences"

Such As…???

• We perceive loudness of sound as the logarithm of the actual sound intensity, and dB (decibels) are a logarithmic scale.

• We also perceive brightness of light as the logarithm of the actual light energy

• Star magnitudes are measured on a logarithmic scale.

• pH or acidity of a chemical solution. The pH is the negative logarithm of the concentration of free hydrogen ions.

• earthquake intensity on the Richter scale is also on a logarithmic scale

What Are Log-Log Plots Used For???

• To analyze exponential processes. • Since the log function is the inverse of

the exponential function, we often analyze an exponential curve by means of logarithms.

• Plotting a set of measured points on "log-log" or "semi-log" scale reveals such relationships clearly

Applications of a Log-Log Plot

• Applications include – cooling of a dead body– growth of bacteria– decay of a radioactive isotopes– the spread of an epidemic in a

population often follows a modified logarithmic curve called a "logistic".

• To solve some forms of area problems in calculus. (The area under the curve 1/x, between x=1 and x=A, equals ln A)