Logarithmic Functions

1
Logarithmic FunctionsApplications of
Logarithmic Functions

• Section 4.3 4.4

2
Objectives

• Understand that logarithmic functions are
  inverses of exponential functions.
• Change from logarithmic to exponential form and
  vice versa.
• Evaluate logarithms.
• Graph logarithmic functions.
• Solve real world applications involving
  logarithms.

3
Earthquakes

• The Richter scale, is used for measuring the
  intensity of an earthquake. For each one unit
  increase on the scale the intensity is 10 times
  greater.
• The earthquake that killed 70 people and injured
  2400 in northern California on October 17, 1989
  measured 7.1 on the Richter scale.
• Logarithmic formula used too measure the
  intensity of an earthquake is

4
Introduction to Logarithms

• y bx is a one-to-one function, therefore, it
  has an inverse.
• To find the inverse switch the x and y and solve
  for y. x by
• To solve for y, we introduce the definition of
  logarithms. (b gt 0 and b ? 1)
• y log b x if and only if x by
• Conclusion
• y log b x is the inverse of y bx

5
Definition of Logarithmic Function

• The inverse function of the exponential function
  with base b is called the logarithmic function
  with base b.
• Definition
• For xgt0 and bgt0, b?1,
• y logbx is equivalent to by x
• The function f(x) logbx is the logarithmic
  function with base b

6
Location of base and exponent in exponential and
logarithmic forms

• Logarithmic form
• y log b x

• Exponential form
• by x

Exponent
Exponent
Base
Base
7
y log b x if and only if x by

• Write each equation in exponential form
• log 3 81 4
• log 7 1/49 -2
• Write each equation in logarithmic form
• 103 1000
• 4-2 1/16

8
Find y in each equation.

• log 2 8 y
• log 5 1 y
• 25

9
Find a in each equation

• log a 36 2
• log 8 a -1/3

10
Graphs of Logarithmic Functions

• Graph
• y log 2 x

11
Graphs of Logarithmic Functions

• Graph
• y log 1/2 x

12
y log b x has the following properties

• Domain (0, ?) , Range (- ?, ?)
• It passes through the point (1,0)
• It passes through the point (b, 1)
• The y- axis is an asymptote.
• If b gt 1, it is an increasing function
• If 0 lt b lt 1 , it is a decreasing function

13
Transformations Involving Logarithmic Functions
14
Translations

• Graph y log 1/3 (x 2)

• Graph y log2 x - 1

15
Base 10 logarithms

• Called common logarithms
• When base b is not indicated, it is understood
  that b 10
• log 1/100 log 10
• log 1/10 log 100
• log 1 log 1000

16
Logarithms involving base -e

• log e x means ln x
• These are called natural logarithms
• y ln x is the inverse of y ex
• The domain of y ln x is (0, ?)
• The range is the interval (-?, ?)

17
Solve each equation
Simplify using a calculator

• ln 17.31
• ln(log 1/3)

• ln x 1.9344
• log x ln 3.2

18
Graph y ln x y ln(3x)
19
Decibel Voltage Gain

• If E0 is the output voltage of a device and EI is
  the input voltage, the decibel voltage gain is
  given by db gain 20 log E0
• EI
• Find the db gain of an amplifier if its input is
  0.7 volt and its output is 40 volts.

20
Richter Scale

• If R is the intensity of an earthquake, A is the
  amplitude (measured in micrometers), and P is the
  period ( the time of one oscillation of the
  earths surface, measured in seconds), then
• Find the intensity of an aftershock with a period
  of 0.07 and an amplitude of 2500 micrometers.

21
Charging Batteries

• If M is the theoretical maximum charge that a
  battery can hold and k is a positive constant
  that depends on the battery and the charger, then
  C M(l e-kt)
• How long will it take to bring a battery to 90
  of full charge? Assume that k 0.025

22
Population Growth

• The time required for the population to double is
  called doubling time.
• If r is the annual rate and t is the time
  required for a population to double, then
• The population of the earth is growing at
  approximately 2 per year. If this rate
  continues when will the population double.

23
Isothermal Expansion

• If the temperature T is constant, the energy E
  required to increase the volume of one mole of
  gas from an initial volume Vi to a final volume
  Vf is given by
• E is measured in joules and T in degrees Kelvin
  and R is the universal gas constant 8.314
  joules/mol/K.

24
Find the amount of energy that must be supplied
to double the volume of 1 mole of gas at a
constant temperature of 300K.