7.3 L OGARITHMIC F UNCTIONS

D ECIBELS Sound is measured in units called decibels. Decibel level begins at zero, which is near total silence. A whisper is at 30 decibels, and a normal conversation is 60 decibels. An increase of 10 means that a sound is 10 times more intense. So, how much more intense is a normal conversation than a whisper?

T RY THIS ONE : An ambulance siren is at 120 decibels. How many more times is that than the weakest sound we can hear (which is at 0)? What is the base we are looking at?

W HY DO WE CARE ? Listening to sounds at high decibel levels for a long period of time can cause hearing loss. Those who are exposed to noise levels at 85 decibels or higher for a prolonged period of time are at risk for hearing loss. MP3’s on loud volumes are 105 decibels. How many more times more intense is that than where the damage will occur?

S O WHAT ? So what does this have to do with these exponential functions we’ve been working with? Parts of an exponential equation: b x = a

T HINK ABOUT THIS : How many times would you have to double $1 before you had $8?

L OGARITHMS A logarithm would help us be able to solve that type of problem, where we’re looking for the exponent. It is the exponent to which a specified base is raised to obtain a given value. An exponential equation and a logarithmic equation are related: ExponentialLogarithmic b x = a log b a = x (means log base b of a is x)

R EWRITE EACH EXPONENTIAL AS A LOGARITHM x0x0

W RITE EACH LOGARITHMIC AS AN EXPONENTIAL log = 2 log ½ 8 = -3 log =?

S PECIAL PROPERTIES OF LOGS For an base b such that b>0 and b ≠ 1: log b b = 1 log b 1 = 0

T HE MOST COMMON LOG A logarithm with base 10 is the common logarithm. This means, if there is no base, then you can assume the base is 10. Example: log 5 means log 10 5 This is the log that is on your calculator.

F IND THE FOLLOWING LOGS USING MENTAL MATH log 1000(think: 10 to what power is 1000?) log 4 ¼(think: 4 to what power is ¼?) log log

S OLVE FOR THE UNKNOWN : log 8 x = 3 log x 343 = 3 log = x