Section 4.4 Properties of Logarithms and Logarithmic Scales
(Recall from Section 4.3)
EXAMPLES:
• Product Property:
• Quotient Property:
• Power Property:
Caution: You cannot distribute "log"!!! It is not a numerical value unless there is the log of something.
Examples of expanding and condensing logarithmic expressions:
We are allowed to take the log of both sides of an equation! (just like we can take the square root of both sides, etc.)
We are also allowed to "drop" the log from both sides of an equation as long as there is a single logarithm with the same base on each side.
So what about 2x = 9??
Change to exponential form? How about if we take the log or ln of both sides?
This leads to the Change of Base Property:
logbx = logax
logab
(Usually we choose the new base "a" to be 10 or e so we can use our calculators.)
Then we have:
Examples:
Applications of Logarithms: (pH, Richter Scale and decibels)
1) The pH of a solution is pH = ­log[H+]
where [H+] is the concentration of hydronium ions in moles/liter. Solutions with pH less than 7 are acidic, while those with a pH greater than 7 are basic.
If orange juice has [H+] of 1.58 x 10­4 moles/liter, what is the pH?
2) Richter Scale Ranking: I0 is the intensity of a just­discernable earthquake and I is the intensity of an earthquake being analyzed:
R = log I I0
(The common log is base = 10 so each increase of 1 unit on the Richter scale is 10 times as intense as a barely felt earthquake)
The January 2001 earthquake in the state of Gujarat in India was 80,000,000 times as intense as a 0­level earthquake. What was the Richter ranking of this devastating event?
3) The intensity of a sound is measured on a decibel scale:
D = 10 log I I0
where I0 is the intensity of a barely discernable sound.
• 0­40 normal conversation
• 41­80 heavy traffic
• 81­120 loud rock concert
• 121­160 human eardrum likely to rapture
Given that I0 = 10­12 watts/meter2, what is the decibel level of a jet airliner's engines at a distance of 45 meters, for which the sound intensity is 50 watts/meter2?
MORE EXAMPLES: