log2

1

8 3

log5 1 0

Write in logarithmic form 38 2 1

3 12

8

Write in logarithmic form 05 1

Write

in exponential form

2

1log 4

16

4 12

16

Write

in exponential form

53log 5 3

3 35 5

Math 30-1 1

When solving a logarithmic equation, try rewriting the

equation in exponential form.

Strategy One: Exponents

Rewrite the problem in exponential form.

62 x

6Solve log 2, 0x x

5y

1

25

Since 1

25 5

2

5

y 5

2

y 2

5

1Solve log

25y

36x

Math 30-1 2

Property of Equality for Logarithmic Equations.

Suppose b 0 and b 1.

Then logb x1 logb x2 if and only if x1 x2

For equations with logarithmic expressions on both sides the equal

sign, if the bases match, then the arguments must be equal.

Strategy Two: Equating Logarithms

7 7log 2 1 log 11x

2 1 11x

2 12x

6x

Restrictions

2 1 0x

1

2x

Math 30-1 3

Strategy Three: Graphically.

Solve: log3(4x10) log3(x1)

Since the bases are both ‘3’ we set the arguments equal.

4x10 x1

3x 101

3x 9

x3

Restrictions

1x

extraneous

No solution

Math 30-1 4

Solving Log Equations

1. log272 = log2x + log212

log272 - log212 = log2x

log2

72

12

log2 x

72

12

x

x = 6

Restrictions

x 0

Math 30-1 5

2.Solve: log8(x214) log8(5x)

x214 5x

x2 5x14 0

(x 7)(x 2) 0

(x 7) 0 or (x 2) 0

x 7 or x 2

Restrictions

extraneous

14

3.7

x

x

Math 30-1 6

3. log7(2x + 2) - log7(x - 1) = log7(x + 1)

log7

2x 2

x 1

log7(x 1)

2x 2

x 1

(x 1)

2x + 2 = (x- 1)(x + 1)

2x + 2 = x2 - 1

0 = x2 - 2x - 3

0 = (x - 3)(x + 1)

x - 3 = 0 or x + 1 = 0

x = 3 x = -1

log7(2x + 2) - log7(x - 1) = log7(x + 1)

log7(2(3) + 2) - log7(3 - 1) = log7(3 + 1)

log74 = log74

log7(2x + 2) - log7(x - 1) = log7(x + 1)

log7(2(-1) + 2) - log7(-1 - 1) = log7(-1 + 1)

log70 - log7(-2) = log7(0)

Negative logarithms and

logs of 0 are undefined.

Therefore, x = 3

Solving Log Equations

Verify Algebraically:

log7

8

2

log7 4

Restrictions

x 1

extraneous

Math 30-1 7

4. log7(x + 1) + log7(x - 5) = 1log7[(x + 1)(x - 5)] = log77

(x + 1)(x - 5) = 7x2 - 4x - 5 = 7x2 - 4x - 12 = 0

(x - 6)(x + 2) = 0x - 6 = 0 or x + 2 = 0

x = 6 x = -2

x = 6

Solving Log EquationsRestrictions

x 5

extraneous

7log 1 50 1x x

17 1 50x x

Math 30-1 8

Solving Exponential Equations Unlike Bases

5. 2x = 8

log 2x = log 8

xlog2 = log 8

x log8

log2

x = 3

6. Solve for x:

2x 12

xlog2 = log12

x log12

log2

x = 3.58

23.58 = 12

Math 30-1 9

Solving Log Equations

8. Solve log5(x - 6) = 1 - log5(x - 2)

log5(x - 6) + log5(x - 2) = 1log5(x - 6)(x - 2) = 1log5(x - 6)(x - 2) = log551

(x - 6)(x - 2) = 5x2 - 8x + 12 = 5x2 - 8x + 7 = 0

(x - 7)(x - 1) = 0x = 7 or x = 1

Since x > 6, the value of x = 1

is extraneous therefore, the

solution is x = 7.

7. 7

1

2x

40

1

2x log7 log40

xlog7 = 2log40

x 2log40

log7

x = 3.79

log 71

2x

log 40

Math 30-1 10

9. 3x = 2x + 1

log(3x) = log(2x + 1)

x log 3 = (x + 1)log 2

x log 3 = x log 2 + 1 log 2

x log 3 - x log 2 = log 2

x(log 3 - log 2) = log 2

x log2

log3 log2

x = 1.71

Solving Log Equations

10. 2(18)x = 6x + 1

log[2(18)x] = log(6x + 1)

log 2 + x log 18 = (x + 1)log 6

log 2 + x log 18 = x log 6 + 1 log 6

x log 18 - x log 6 = log 6 - log 2

x(log 18 - log 6) = log 6 - log 2

x log6 log2

log18 log6

x =1

Math 30-1 11

Page 4121, 2, 3, 4b,c, 5a,c, 6, 7c,d, 8, 18

Math 30-1 12

Logarithms can be used in measuring quantities which vary widely.

We use the log function because it “counts” the number of powers of 10. This is necessary because of the vast range of some physical quantities we measure.

Exponential growth or decayLoans, mortgages, investmentsUsing a log scale:Sound intensity (decibels)Acidity (pH) of a solutionEarthquake intensity (Richter scale)

Math 30-1 13

Algebraically determine the time period required for $7000

invested at 10%per year compounded semiannually to grow to

$10 000.

10 000 = 7000(1.05)2t

log10 log 72

log1.05t

7.31 = 2t

3.66 = t

It would take 4 years for the investment to grow to $10 000.

Compound Interest

log10 - log7 = 2tlog1.05

10log 2 log1.05

7t

2101.05

7

t

Final Initial Factor total time

time for 1 period

1/20.1010000 7000 1

2

t

Math 30-1 14

A biologist originally estimates the number of E. coli bacteria in a

culture to be 1000. After 90 min, the estimated count is 19 500

bacteria. What is the doubling period of the E. coli bacteria, to the

nearest minute?

( )

t

py a b90

19500 1000(2) p90

19.5 (2) p

p = 21

The doubling period is approximately 21 minutes.

Exponential Growth

Final Initial Factor total time

time for 1 period

90

log19.5 log(2) p

90log19.5 log(2)

p

90log 2

log19.5p

19.590log 2p

Math 30-1 15

For every metre below the water surface, light intensity is

reduced by 5%. At what depth, to the nearest hundredth of a

metre, is light intensity 40% of that at the surface?

0.40 = 1(1 - 0.05)d

0.40 = 1(0.95)d

0.4 = 0.95d

log 0.4 = dlog0.95

d log0.4

log0.95

d = 17.86

Therefore, at a depth of 17.86 m

the light intensity would be 40%.

Light Intensity

t

py a b

0.95log 0.4d

Math 30-1 16

Decibels

Suppose I0is the softest sound the human ear can hear, measured in watts/cm2

I is the watts/cm2 of a given sound

0

10 logdB

IL

I

The log of the

ratio

Then the decibels of the sound is

Math 30-1 17

Comparing Intensities of Sound

For any intensity, I, the decibel level, dB, is defined as follows:

LdB 10logI

Io

where Io is the intensity of a barely

audible sound

The sound at a rock concert is 106 dB. During the break the

sound is 76 dB. How many times as loud is it when the band is

playing?

Louder

LdB 10logI

Io

106 10logI

Io

10.6 logI

Io

I

Io 10

10.6

I = 1010.6 Io

Softer

LdB 10logI

Io

76 10logI

Io

7.6 logI

Io

I

Io 10

7.6

I = 107.6 Io

Ilouder

Isofter

1010.6 Io

107.6Io

Ilouder

Isofter 10

3

106 dB would be

1000 times as loud as

76 dB.

Comparison

Math 30-1 18

pH scale to measure acidity or alkalinity of a solution

pH log H /mol L

A common ingredient in cola drinks is phosphoric acid, the same ingredient found in many rust removers.

A cola drink has a pH of 2.5, milk is 6.6. How many times as acidic as milk is a cola drink?

Cola

2.5 log H

Milk

2.5

6.6

10

10

Cola

Milk

Comparison6.6 log H

2.510 H 6.610 H

4.1Cola

MilkCola would be 4.1 times

as acidic as Milk.Math 30-1 19

Measuring Earthquakes

Seismic wavesradiated by all earthquakes can provide good estimates of their magnitudes

Math 30-1 20

Comparable Magnitudes

Richter TNT for Seismic ExampleMagnitude Energy Yield (approximate)

• -1.5 6 ounces Breaking a rock on a lab table• 1.0 30 pounds Large Blast at a Construction Site• 1.5 320 pounds• 2.0 1 ton Large Quarry or Mine Blast• 2.5 4.6 tons• 3.0 29 tons• 3.5 73 tons • 4.0 1,000 tons Small Nuclear Weapon• 4.5 5,100 tons Average Tornado (total energy)• 5.0 32,000 tons• 5.5 80,000 tons Little Skull Mtn., NV Quake, 1992• 6.0 1 million tons Double Spring Flat, NV Quake, 1994• 6.5 5 million tons Northridge, CA Quake, 1994• 7.0 32 million tons Hyogo-Ken Nanbu, Japan Quake, 1995;

Largest Thermonuclear Weapon• 7.5 160 million tons Landers, CA Quake, 1992• 8.0 1 billion tons San Francisco, CA Quake, 1906• 8.5 5 billion tons Anchorage, AK Quake, 1964• 9.0 32 billion tons Chilean Quake, 1960• 10.0 1 trillion tons (San-Andreas type fault circling Earth)• 12.0 160 trillion tons (Fault Earth in half through center,

OR Earth's daily receipt of solar energy)

http://earthquake.usgs.gov/earthquakes/map/

Math 30-1 21

Applications - The Richter Scale

I = Io(10)m where m is the measure on the scale (magnitude)

Compare the intensities of the Japan earthquake of 1933,

which measured 8.9 on the Richter Scale, to the earthquake

of Turkey in 1966, which measured 6.9 on the scale.

IJapan

ITurkey

108.9 Io

106.9Io

IJapan

ITurkey 10

2

Therefore, the earthquake in Japan is

100 times as intense as the one in Turkey.

Math 30-1 22

Applications - The Richter Scale

The magnitude of earthquakes is given by m logI

Io

where I is the quake intensity and Io is the reference

intensity.

How many times as intense is a quake of 8.1

compared to a quake with a magnitude of 6.4?

m logI

Io

8.1 logI1

Io

108.1I1

Io

6.4 logI2

Io

106.4I2

Io

Comparison

108.1I1

Io

106.4I2

Io

101.7I1

I2

Therefore, a quake of 8.1 is 50.1 times as great.

I = Io(10)m

Math 30-1 23

Math 30-1 24

Textbook p. 412 – 415

Level 1: (Basic Drill and Practice)

1 – 5, 7, 8

Level 2: (Problem Solving)

9, 10, 11, 12, 13, 14

Level 3: (Extension and Higher Level)

6, 15, 16, 17, 18, 19, 20, 21, 22, C1