Section 4.3The Logarithmic Function

• Recall the definition of log

• In your groups see if you can draw the graph of log (without the aid of a calculator!!!)

• Here are some hints– How are the graphs of y = log x and x = 10y related?

– If x = 1, then what dows the equation x = 10y say that y has to be?

– If x < 1. then according to the equation x = 10y , what has to happen to y?

– If x > 1. then according to the equation x = 10y , what has to happen to y?

– If x is close to zero, then 10y has to be close to zero also. What does y have to do in order to get 10y close to zero?

xxy y 10thenlogIf

• Let’s look at the graph of y = log x together– What is its vertical intercept?

• None

– What is its horizontal intercept?• (1, 0)

– Does it have a horizontal asymptote?• No, it continues increasing without bound

– Does it have a vertical asymptote?• Yes, the line x = 0

• Recall:– The graph has a horizontal asymptote at y = a if

the function approaches the value of a as x goes to positive or negative infinity

– The graph has a vertical asymptote at x = a if the function goes to positive or negative infinity as x approaches a (from the right, left or both)

• So what about the graph of y = ln x?– What is its vertical intercept?

• None

– What is its horizontal intercept?• (1, 0)

– Does it have a horizontal asymptote?• No, it continues increasing without bound

– Does it have a vertical asymptote?• Yes, the line x = 0

• In fact, y = 10x = (eln(10))x so the two graphs differ by a factor of ln(10)

Applications of Logs

• Chemical Acidity (pH level) The acidity depends on the hydrogen ion concentration in

the liquid (in moles per liter) written [H+]. The greater the hydrogen ion concentration, the more acidic the solution

The pH is defined as pH = -log [H+] Pure water contains a hydrogen ion concentration of 1 x

10-7 moles. What is its pH?

Some facts about pH• pH is used to measure the acidity of a liquid• pH stands for “power of hydrogen” or “potential

hydrogen”• pH is a critical measurement. Life depends upon it.

For instance, human blood is basic with a pH between 7.3 and 7.5. If the pH of blood drops below 7.3, acidosis occurs. If the blood pH rises above 7.5, alkalosis occurs. Death will occur if blood pH goes below 7.0 or above 7.8. Our human existence depends upon a balanced and buffered blood pH.

• Sound intensity is measured in orders of 10– These are known as orders of magnitude– For example, the softest audible sound intensity is

approximately 10-16 watts/cm2 (this is often denoted Io). The sound intensity of a refrigerator motor is 10-11 watts/cm2. Compare the two intensities.

• We do this by comparing the ratio of the two intensities

– The intensity of audible sounds has a very wide range so to scale down the numbers we use logarithm of the sound intensity, measured in decibels (dB)

– Noise levels in decibels =

0

log10I

I

Example• A typical high school cafeteria has a sound

intensity of 10-8 watts/cm2

– How many decibels is this? • Recall that Io = 10-16 watts/cm2 and dB =

• Now suppose we have one sound measured at 65 dB and another measured at 50 dB. How many times more intense is the first sound than the second?– Hint: Create an equation for each intensity and use

properties of logs

0

log10I

I

• In the time remaining work on the following problems in your groups: 7, 28, and 32