Logarithmic Graphs:

Understanding the Characteristics and Behavior of Logarithmic Graphs

Lesson Overview

Instructor Presentation

Introduction

Logarithmic functions follow the form y = (Subscript[log, a])x with the following definitions:

a = base of the function, a > 0 and a ≠ 1.

x > 0

The function is always increasing or always decreasing.

The function is continuous (no breaks, holes, or gaps in the graph) everywhere

Plot[Log[2,x],{x,0.001,5}, PlotRange->{-10,2},PlotStyle->{Thick, Blue}]

Goals

Students will be able to plot and analyze graphs of logarithmic functions of the form f(x) = d + (Subscript[log, a])(x - c) using Mathematica.

parameters ={a,c,d,x}
{a,c,d,x}
definition = {base,horizontal shift, vertical shift, independent variable}
{base,horizontal shift,shift vertical,independent variable}
Grid[Join[{{Style["parameters",FontFamily->"Arial",Bold],Style["definition",FontFamily->"Arial",Bold]}},
Partition[Riffle[parameters,definition],2]],Dividers->{2->True,2->True}]

The value of x must be greater than 0, since there is no exponent (y) such that raising a positive number to that exponent would yield a negative number.  Hence, the graph of  y = (Subscript[log, a])x contains a vertical asymptote at x = 0 (the y-axis).

Students will make correct conjectures about the appearance and behavior of logarithmic graphs after transformations are applied.

Plot[Log[3,x],{x,-2,12},PlotRange->{-5,4},PlotStyle->{Thick,Blue}]

In logarithmic functions, in general, the larger the base, the less steep the graph appears.  Although, this is hard to determine with nearly any logarithmic graph, especially for  x > a.

Because there is no limit to the value of x, and that no value of x > 0 would render the function undefined, y will increase without bound.  Hence, the graph of  contains no horizontal asymptotes.

Horizontal and vertical shifting occurs when "c" and "d" are non-zero numbers.  This shifting follows the same rules as in polynomial functions, studied earlier.

Students will answer real-world problems by applying or using logarithms.

Logarithmic models are used in real-world situations, such as loudness of sound and earthquake monitoring.

Concept Exploration

Bases of Logarithmic functions

Manipulate[Plot[Log[a,x],{x,-1,8}, PlotStyle->{Thick,Blue},PlotRange->{-6,6}, PlotLabel->Grid[{{"y =",(Subscript[log, a])x}}]],{a,1.0001,5}]

For values of the base between 0 and 1, the graph appears to be reflected in the x-axis.  As x approaches 0 from the right, the graph is "compressed" more toward the x-axis.

For values of the base greater than 1, as the base increases, the same effect happens - the graph is "compressed" or "hugs" the x-axis closer.  .

As the base approaches 1 from the right or left, the graph trends toward the vertical line x = 1.

Logarithmic functions - Shifting and Reflecting

Plot[{Tooltip[Log[2,x],"(Subscript[log, 2])x"],Tooltip[Log[2,x]+3,"3 + (Subscript[log, 2])x"],Tooltip[ Log[2,x+4],"(Subscript[log, 2])(x + 4)"]}, {x,-10,10}, PlotRange->{-10,10},PlotStyle->{{Thick,Blue},{Thick,Red},{Thick,Green}}]

Vertical and horizontal shifting behave in the same way for logarithmic graphs as with other "algebraic" graphs previously studied.

Logarithmic functions and Reflection

Manipulate[Show[Plot[{Log[a,x],Log[a,-x]}, {x,-10,10}, PlotRange->{-10,10},PlotStyle->{{Thick,Green},{Thick,Purple}}],PlotLabel->Grid[{{"y =",Log[a,x]},{"y =",Log[a,-x]}}]],{a,1.01,5}]

If we multiply the x by -1, the graph reflects across the y-axis.

Manipulate[Show[Plot[{Log[a,x],-Log[a,x]}, {x,-10,10}, PlotStyle->{{Thick,Blue},{Thick,Red}},PlotRange->{-10,10}],PlotLabel->Grid[{{"y =",Log[a,x]},{"y =",-Log[a,x]}}]],{a,1.01,5}]

If we multiply the ENTIRE function by -1, the graph reflects across the x-axis.

Examples - Logarithmic Graphs

Example 1

Run the following demonstration and give an explanation for what each variable does to the graph when changed.

Manipulate[Show[Plot[d*Log[a,x+b]+c,{x,-10,10},PlotStyle->{Thick,Orange},PlotRange->{-10,10}],PlotLabel->d*Log[a,x+b]+c],{a,0.1,5},{b,-5,5},{c,-5,5},{d,{-1,1}}]

a :

b :

c :

d :

Example 2

Create and plot the following exponential graphs choose an appropriate window.

An logarithmic graph with a base of 12, reflected in the x-axis.

An logarithmic graph with a base of 2, shifted down 9 units and left 1 units

An logarithmic graph with a base of 8, shifted down 3 unit, then reflected in the y-axis.

Example 3

Describe the transformations, in the correct order, that occurred to create the given exponential graph.

y = (Subscript[log, 4])(x+5)

y = 5 - (Subscript[log, 2.6])3

y = -(Subscript[log, 9])(x-1) + 8

Concept Exploration

Logarithmic Models

There are many applications to logarithms in real-life examples.  One instance is the "loudness of sound" measurement.  Look at the table below...

sound={Hearing threshold," Soft whisper at 5 m", City traffic, Subway train, Pain threshold,"Jet at takeoff"}

intensity = {"(10^(-12))", "(10^(-11))", "(10^(-5))", "(10^(-2))", "(10^0)", "(10^3)"}

Grid[Join[{{Style["Sound",FontFamily->"Arial",Bold],Style["Intensity (W/(m^2))",FontFamily->"Arial",Bold]}},
Partition[Riffle[sound,intensity],2]],Dividers->{2->True,2->True}]

The level of sound intensity in decibels (dB) is

({{({{β = 10 log ((I/(Subscript[I, 0])))}})}})

where β is the number of decibels, I is the sound intensity in W/(m^2) and (Subscript[I, 0]) = (10^(-12)) W/(m^2) is the threshold of human hearing (the quietest audible sound intensity).

How loud is a train inside a subway tunnel?  (Look at table above).

({{({{β = 10 log((I/(Subscript[I, 0])))}, {β = 10 log (((10^(-2))/(10^(-12))))}, {β = 10 log ((10^10))}, {β = 10 · 10 = 100 dB}})}})

Example - Logarithmic Models

Example 4 - Intensity of Light

The Beer-Lambert law of absorption applied to Lake Erie states that the light intensity I (in lumens), at a depth of x feet, satisfies the equation

({{log(I/12) = -0.00235x}})

Find the intensity of the light at depths of 10 ft, 20 ft, and 30 ft.

Example - Logarithmic Models

Example 5 - Earthquakes

The Richter Scale magnitude R of an earthquake is based on the features of the associated seismic wave and is measured by

({{({{R = log((a/T)) + B}})}})

where a is the amplitude in μm (micrometers), T is the period in seconds, and B accounts for the weakening of the seismic wave due to the distance from the epicenter.  Use this equation to find out how many times more severe the August 16, 2009 Indonesian earthquake ((Subscript[R, 1]) = 6.6) was than the one earlier that day near the Dominican Republic ((Subscript[R, 2]) = 3.1)?

Adapted from Precalculus: Graphical, Numerical, Algebraic, 6th edition, Demana, Waits, Foley, Kennedy and http://earthquake.usgs.gov/eqcenter/recenteqsww/Quakes/quakes_all.html

Examples Answer Key

Example 1

a : As a approaches + ∞, the graph "hugs" the x - axis.  As a approaches 1 from the right or left, the graph trends toward a vertical line at x = 1.  Between the values of 0 and 1, the graph appears to be inverted about the x - axis.  As a approaches 0 from the right, the graph "hugs" the x - axis again.

b : Indicates the horizontal shift.  As b gets smaller, the graph shifts right b units.  As b gets larger, the graph shifts left b units.

c : Indicates the vertical shift.  As c increases, the graph shift up c units.    As c decreases, the graph shifts down c units.

d : When d = -1, the graph is decreasing.  When d = 1, the graph is increasing.  Comparing the same graph, except the value of d, when changing from 1 to - 1, the graph is reflected about a horizontal line, namely, y = c.

Example 2

A logarithmic graph with a base of 3, reflected in the x-axis.  Graph in an appropriate window.

Plot[-Log[3,x], {x,-2,10},PlotStyle->Thick]

A logarithmic graph with a base of 1.5, shifted up 4 units and right 1 unit.

Plot[Log[1.5,(x-1)]+4,{x,-2,10},PlotRange->{-10,10},PlotStyle->Thick]

A logarithmic graph with a base of (1/3) , shifted down 4 units, then reflected in the y-axis.

Plot[-4+Log[(1/3),-x],{x,-10,10},PlotRange->{-10,4},PlotStyle->Thick]

Example 3

Describe the transformations, in the correct order, that occurred to create the given logarithmic graph.

y = -(Subscript[log, 3])x + 8

Logarithmic graph with a base of 3.  Reflected in the x - axis and shifted 
up 8 units.

y = (Subscript[log, 2])(x+14)

Logarithmic graph with a base of 2.  Shifted left 14 units.

y = -3 - (Subscript[log, 9])(x-30)

Logarithmic graph with a base of 9.  Shifted right 30 units, then reflected
in the x - axis, then shifted down 3 units.

Example 4 -

The Beer-Lambert law of absorption applied to Lake Erie states that the light intensity I (in lumens), at a depth of x feet, satisfies the equation

({{log(I/12) = -0.00235x}})

Find the intensity of the light at depths of 10 ft, 20 ft, and 30 ft.

Solve[Log[10,a/12] == -0.00235*10,a]

Solve[Log[10,a/12] == -0.00235*20,a]

Solve[Log[10,a/12] == -0.00235*30,a]

The light intensity at 10ft, 20 ft, and 30 ft is 11.37 lumens, 10.77 lumens, and 10.20 lumens, respectively.

Example 5 - Earthquakes

The Richter Scale magnitude R of an earthquake is based on the features of the associated seismic wave and is measured by

({{({{R = log((a/T)) + B}})}})

where a is the amplitude in μm (micrometers), T is the period in seconds, and B accounts for the weakening of the seismic wave due to the distance from the epicenter.  Use this equation to find out how many times more severe the August 16, 2009 Indonesian earthquake ((Subscript[R, 1]) = 6.6) was than the one earlier that day near the Dominican Republic ((Subscript[R, 2]) = 3.1)?

({{({{(Subscript[R, 1]) = log(((Subscript[a, 1])/T))}}) + B = 6.6}, {(Subscript[R, 2]) = log(((Subscript[a, 2])/T)) + B = 3.1}})

Algebraically, we seek the ratio of severities ((Subscript[a, 1])/(Subscript[a, 2])):

({{({{({{(log(((Subscript[a, 1])/T)) + B) - (log(((Subscript[a, 2])/T)) + B)}}) = (Subscript[R, 1]) - (Subscript[R, 2])}, { log(((Subscript[a, 1])/T)) - log(((Subscript[a, 2])/T)) = 6.6 - 3.1     (B - B = 0)}, {log(((Subscript[a, 1])/(Subscript[a, 2]))) = 3.5                      (Quotient Rule of Logarithms)}, {((Subscript[a, 1])/(Subscript[a, 2])) = (10^3.5) ≈ 3,162.28}})}})

The Indonesian earthquake was 3,162 times more powerful than the Dominican Republic earthquake.