Logarithms

1
Chapter 2 Exponents and Logarithms
2.7A
Logarithms and Equation Solving
2.7A.1
MATHPOWERTM 12, WESTERN EDITION
2
Solving Log Equations
1. log272 log2x log212
log272 - log212 log2x
2. 2x 8 log 2x log 8 xlog2
log 8
x 3
x 6
3.
x 3.79
xlog7 2log40
2.7A.2
3
Solving Log Equations
4. log7(2x 2) - log7(x - 1) log7(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
Therefore, x 3.
Check
log7(2x 2) - log7(x - 1) log7(x 1)
log7(2x 2) - log7(x - 1) log7(x 1)
log7(2(-1) 2) - log7(-1 - 1) log7(-1 1)
log7(2(3) 2) - log7(3 - 1) log7(3 1)
log70 - log7(-2) log7(0)
Negative logarithms and logs of 0 are undefined.
log74 log74
2.7A.3
4
Solving Log Equations
log7(x 1) log7(x - 5) 1 log7(x
1)(x - 5) log77 (x 1)(x -
5) 7 x2 - 4x - 5 7
x2 - 4x - 12 0
(x - 6)(x 2) 0 x - 6 0 or
x 2 0 x 6
x -2
6. log4(4x) - log4(x - 2) 3
5.
64(x - 2) 4x 64x - 128 4x 60x
128 x 2.13
x -2 is extraneous. Therefore, x 6.
2.7A.4
5
Solving Log Equations
Solve log5(x - 6) 1 - log5(x - 2)
Express 12 as a power of 2
log5(x - 6) log5(x - 2) 1 log5(x -
6)(x - 2) 1 log5(x - 6)(x - 2)
log551 (x - 6)(x - 2) 5
x2 - 8x 12 5
x2 - 8x 7 0 (x - 7)(x - 1)
0 x 7 or x 1
xlog2 log12
x 3.58
23.58 12
Since x gt 6, the value of x 1 is extraneous.
Therefore, the solution is x 7.
2.7A.5
6
Solving Log Equations
7. 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
8. 23x - 1 32x - 1
(3x - 1) log 2 (2x - 1) log 3
3x log 2 -1 log2 2x log3 - log3 3xlog2
- 2xlog3 log2 - log3 x(3log2 - 2log3)
log2 - log3
x 3.44
x 1.71
2.7A.6
7
Applications of Logarithms
1. Carbon 14 has a half-life of 5760 years. Find
the age of a specimen with 24 C-14
relative to living matter.
t 11 859.23
Therefore, the specimen is 11 859.23 years old.
2.7A.7
8
Applications of Logarithms

• Find the time period required for 7000 invested
  at
• 10/a compounded semi-annually to grow to 10
  000.

A(t) P(1 i) 2n 10 000 7000(1.05)2n
log10 - log7 2nlog1.05
7.31 2n 3.66 n
It would take 3.66 years for the investment to
grow to 10 000.
2.7A.8
9
Applications of Logarithms
3. The value of an investment is given by f(x)
237.50(1.052)x, where x is the number of
6-month periods. Find the number of
complete periods until the investment is
worth at least 600.
f(x) 237.50(1.052)x 600
237.50(1.052)x 2.5263 (1.052)x log 2.5263
x log 1.052
x 18.28
Therefore, after 19 periods the investment would
be worth at least 600.
2.7A.9
10
Applications of Logarithms
4. Cell population doubles every 3 h. How long
would it take 4 cells to reach a count of
16 384?
It would take 36 h to reach 16 384 cells.
36 t
2.7A.10
11
Applications of Logarithms
5. For every metre below the water surface,
light intensity is reduced by 5. At what
depth is light intensity 40 of that at the
surface?
Id Io(1 - 0.05)d 40
100(0.95)d 0.4 0.95d log 0.4 dlog0.95

d 17.86
Therefore, at a depth of 17.86 m the light
intensity would be 40.
2.7A.11
12
More Applications - Comparing Intensities of Sound
For any intensity, I, the decibel level, dB, is
defined as follows
where Io is the intensity of a barely audible
sound
6. 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?
Comparison
Louder
Softer
Thus, it would be 1000 times as loud.
I 107.6 Io
I 1010.6 Io
2.7A.12
13
More Applications - The Richter Scale
I Io(10)m
where m is the measure on the scale
7. 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.
Therefore, the earthquake in Japan is 100 times
as intense as the one in Turkey.
2.7A.13
14
More Applications - The Richter Scale
8. The magnitude of earthquakes is given by
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?
Comparison
Therefore, a quake of 8.1 is 50.1 times as great.
2.7A.14
15
More Applications - The Richter Scale
9. Earthquake intensity is given by I Io
(10)m, where m is the magnitude and Io is
the relative intensity. A quake of
magnitude 7.9 is 120 times as intense as a
tremor. What is the magnitude of the tremor?
Iq Io (10)7.9
It Io (10)m
log 120 (7.9 - m) log 10
log 120 (7.9 - m)
The magnitude of the tremor is 5.8.
m 7.9 - log 120 m 5.8
2.7A.15
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Assignment
Suggested Questions Pages 113 and 114 A 1-13
odd, 21-25, 40 a-e B 26-33, 40 f-h,
42-44 Page 99 46
2.7A.16
17
Chapter 2 Exponents and Logarithms
2.7B
Logarithmic Identities
2.7B.17
MATHPOWERTM 12, WESTERN EDITION
18
Proving Identities
Some equations are solved for all values of the
variable for Which both sides of the equation
are defined. These equations are called
identities.
logax -1 -1logax
-logax
-logax
L.S. R.S. Therefore, the identity is
proved.
L.S. R.S.
2.7B.18
19
Proving Identities
Prove

Conclusion This suggests that
L.S. R.S.
2.7B.19
20
Proving Identities
Verify
L.S. R.S.
2.7B.20
21
Assignment
Suggested Questions Page 106 44-46 Page 113 33-38
2.7B.21