Warm Up1. How does the function f(x) = log (x-3)+4 differ from the parent function?

2. Compute a. log (4 10)b. log (4 102)c. log (4 103)d. log (4 104)

3. Solve for x (with no calculator):

a. 1.602

b. 2.602

c. 3.602

d. 4.602

Moved up 4, moved right 3,

x= ln 5

Orders of Magnitude

Can be used to compare any like quantities:•A kilometer is 3 orders of magnitude longer than a meter.

•A dollar is 2 orders of magnitude greater than a penny.

•A horse weighting 400kg is 4 orders of magnitude heavier than a mouse weighing 40g.

Logarithms and Orders of Magnitude

•We use the log function because it “counts” the number of powers of 10

•This is necessary because of the vast range of some physical quantities we must measure• Sound intensity• Earthquake intensity

Decibels

Suppose I0 is the softest sound the human ear can hear

• measured in watts/cm2

And I is the watts/cm2 of a given sound

Then the decibels of the sound is

0

10 logI

I

The log of the ratio

The log of the ratio

Decibels

ApproxDecibel

Level Example

0 Faintest sound heard by human ear.

30 Whisper, quiet library.

60 Normal conversation, sewing machine, typewriter.

90 Lawnmower, shop tools, truck traffic; 8 hours per day is the maximum exposure to protect 90% of people.

100 Chainsaw, pneumatic drill, snowmobile; 2 hours per day is the maximum exposure without protection.

115 Sandblasting, loud rock concert, auto horn; 15 minutes per day is the maximum exposure without protection.

140 Gun muzzle blast, jet engine; noise causes pain and even brief exposure injures unprotected ears. Maximum allowed noise with hearing protectors.

Orders of Magnitude

• Sound or noise, the bel, measures the order of magnitude of its intensity compared to the threshold of hearing.

• A sound of 3 bels or 30 dB (decibels) has a sound intensity 3 orders of magnitude above the threshold of hearing.

I = intensity of soundI0= smallest audible sound 10-16 watts/cm2

Earthquakes• The Richter scale measures the intensity of an earthquake as a measure of

the power or energy involved.

• I = Intensity of the earthquake (a measure of the wave energy)• I0 = the minimum intensity used for comparison (assume 1)

Comparing Earthquake intensities

1. How many times more severe was the 2001 earthquake in Gujarat, India (R1=7.9) than the 1999 earthquake in Athens, Greece (R2=5.9)?

We want the ratio of severities I1/I2

9.51

log 22

I

R

Comparing Earthquake intensities

A Richter scale difference of 2 corresponds to an amplitude ratio of 2 powers of 10, or 100. So the Gujarat quake was 100 times as severe as the Athens quake.

Slide 3- 12

pH

In chemistry, the acidity of a water-based solution is measured by the concentration of hydrogen ions in the solution (in moles per liter). The hydrogen-ion concentration is written [H+]. The measure of acidity used is pH, the opposite of the common log of the hydrogen-ion concentration: pH = -log [H+]More acidic solutions have higher hydrogen-ion concentrations and lower pH values.

Comparing Chemical AciditySome especially sour vinegar has a pH of 2.4, and a box of Leg and Sickle baking soda has a pH of 8.4.a) What are their hydrogen-ion concentrations?b) How many times greater is the hydrogen-ion concentration of the

vinegar that that of the baking soda?c) By how many orders of magnitude do the concentrations differ?

Comparing Chemical Aciditya) Vinegar:

Baking Soda:

literper moles 1098.310 34.2 xH

4.8log H

b) How many times greater is the hydrogen-ion concentration of the vinegar that that of the baking soda?

c) By how many orders of magnitude do the concentrations differ?

The hydrogen-ion concentration of the vinegar is 6 orders of magnitude greater than that of the baking soda, exactly the difference in their pH values

Slide 3- 16

Newton’s Law of Cooling

0

An object that has been heated will cool to the temperature of the medium in

which it is placed. The temperature of the object at time can be modeled by

( ) ( ) for an appropriate vakt

m m

T t

T t T T T e

0

lue of , where

the temperature of the surrounding medium,

the temperature of the object.

This model assumes that the surrounding medium maintains a constant

temperature.

m

k

T

T

A hard-boiled egg at temperature 96°C is placed in 16°C water to cool. Four minutes later the temperature of the egg is 45°C. Use Newton’s Law of Cooling to determine when the egg will be 20°C.

T0=96°C

Tm=16°C

We know that T=45 when t=4

We want to find out t when T=20°C