Logarithms: Common and Natural Logs, Calculations, and Applications, Slides of Algebra

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§9.3b

Base 10 & e Logs

Review §

 Any QUESTIONS About

• §9.3 → Introduction to Logarithms

 Any QUESTIONS About HomeWork

• §9.3 → HW-

9.3 MTH 55

Common Log Convention

• By this Mathematics CONVENTION the abbreviation log, with no base written, is understood to mean logarithm base 10, or a common logarithm. Thus, log21 = log 1021
• On most calculators, the key for common logarithms is marked

LOG

Example  Calc Common Log

• Use a calculator to approximate each common logarithm. Round to the nearest thousandth if necessary. a. log(456) b. log(0.00257)
• Solution by Calculator LOG key
  • log(456) ≈ 2.659 → 10 2.659^ = 456
  • log(0.00257) ≈ −2.5901 → 10 −2.5901^ =

Example  Sound Intensity

• This function is sometimes used to calculate sound intensity

10log

I d I

  = (^)    

 Where

• d ≡ the intensity in decibels,
• I ≡ the intensity watts per unit of area
• I 0 ≡ the faintest audible sound to the average human ear, which is 10−^12 watts per square meter (1x10−^12 W/m 2 ).

Example  Sound Intensity

• Use the Sound Intensity Equation (a.k.a. the “dBA” Eqn) to find the intensity level of sounds at a decibel level of 75 dB?
• Solution: We need to isolate the intensity, I , in the dBA eqn (^0)

10log

I d I

  = (^)    

Example  Sound Intensity

• Thus the Sound Intensity at 75 dB is 10 −4.5^ W/m^2 = 10 −9/2^ W/m^2
• Using a Scientific calculator and find that I = 3.162x10− W/m^2 = 31.6 μW/m^2

Example  Sound Intensity

• Check If the sound intensity is 10−4.5^ W/m^2 , verify that the decibel reading is 75.4. 12

10 10log 10

d

− −

  = (^)     d =10log10^ 7.

d =10 7.5 ( )

d = (^75) 

Graph log by Translation

• Reflect in x-axis

y = − log (^) ( x − (^2) )

 Shift UP 2 units

y = 2 − log (^) ( x − (^2) )

Example  Total Recall

• The function P = 95 – 99 ∙log x models the percent, P , of students who recall the important features of a classroom lecture over time, where x is the number of days that have elapsed since the lecture was given.
• What percent of the students recall the important features of a lecture 8 days after it was given?

Natural Logarithms

• Logarithms to the base “ e” are called natural logarithms, or Napierian logarithms, in honor of John Napier, who first “discovered” logarithms.
• The abbreviation “ln” is generally used with natural logarithms. Thus, ln 21 = log e 21.
• On most calculators, the key for natural logarithms is marked

LN

Natural Logarithms

• The logarithm with base e is called the natural logarithm and is denoted by ln x. That is, ln x = log e x. So y = ln x if and only if x = e y
• Applying the basic properties of logs
  1. ln( e ) = 1
  2. ln(1) = 0
  3. ln( ex ) = x
  4. e ln x^ = x

Example  Compound Interest

• In a Bank Account that Compounds CONTINUOUSLY the relationship between the $-Principal, P , deposited, the Interest rate, r , the Compounding time-period, t , and the $- Amount, A , in the Account:

1 ln

A t r P

=

Example  Compound Interest

• If an account pays 8% annual interest, compounded continuously, how long will it take a deposit of $25,000 to produce an account balance of $100,000?
• Familiarize In the Compounding Eqn replace P with 25,000, r with 0.08, A with $100,000, and then simplify.