Transcript Review of Logs - George Mason University

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)