6
◙ EP-Program
- Strisuksa School - Roi-et
Math : Exponential and Logarithmic Function
► Dr.Wattana Toutip - Department of Mathematics – Khon Kaen University
© 2010 :Wattana Toutip
◙ [email protected]
◙ http://home.kku.ac.th/wattou
6 Exponential and Logarithmic Functions
6.1 Laws of indices and logarithms
Indices obey the following laws:
a n  a m  a nm
a n  a m  a nm
a 
n m
 a nm
an 
a m/ n  n a m 
1
an
 a
n
m
The exponential function is defined as y  a x where a  0 and a  1 .
The logarithm function is the inverse of the exponential function. i.e.:
a y  x if and only if y  log a x .
This is read : log to the base a of x . If the base is not mentioned then by convention it is 10 .
log x  log10 x .
Logarithms to the base e (approximately equal to 2.71828... ) are called natural logarithms.
They are written as ln x .
ln x  log e x
Logarithms obey the following laws:
1) log a x  log a y  log a xy
2) log a x  log a y  log a xy
◙ EP .Program – Strisuksa School Roi-et.
Mathematics
6. Exponential and Logarithmic Functions. page 2
3) log a x  log a y  log a x / y
4) loga xn  n loga x
To change the base of logarithm use the following:
log a x
5) log b x 
log a b
In particular,
log x
6) log a x 
log a
To convert to powers of 10 or of e use the following:
7) a x  10 x log a  e x ln a
6.1.1 Examples
1. Simplify the expression 2 log 2 x  log 2 3 by writing it as a single log .
Solution
First write 2 log 2 x as log 2 x2 .Then use the rule for addition of logs .
2log2 x  log 2 3  log 2 x2  log 2 3
 log2 ( x2  3)
2log2 x  log2 3  log 2 (3x2 )
2. Solve the equation  e x   2e x  15  0
2
Solution
e2 x   e x  .Write the equation as a quadratic in e x .
2
 e   2e
 e  5 e
x 2
x
 15  0
x
x
 3  0
e x  5 or 3 .
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◙ EP .Program – Strisuksa School Roi-et.
Mathematics
6. Exponential and Logarithmic Functions. page 3
e x  5 has no root.
The solution for e x  3 is : x  ln 3  1.0986
■
3. The population of a country is increasing at 3% per annual. At this rate of growth
how long will it before the population doubles?
Solution
103
Every year the population will be multiplied by
 1.03 .
100
After x years the population will be multiplied by 1.03x .If it has doubled after x
year, this gives the equation:
2  1.03x
Take logs of both sides: log 2  log1.03x  x log1.03
log 2
 23.45
log1.03
The population doubles after 23.45 years
x
6.1.2 Exercises
1. Without use of a calculator evaluate the following:
(a) 161/2
(b) 1003/2
1
(c)  
4
2
(d) 84/3
(e) 5  251/4 1251/6
(f) 31/2  91/4
2. Simplify the following as far as possible:
(a) x3  x 2  x5
(b) y1/2  y 3/2  y 1
(c) 5x  25x 1252 x
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◙ EP .Program – Strisuksa School Roi-et.
Mathematics
6. Exponential and Logarithmic Functions. page 4
(d) 22 n  43n 163n
3. Write the following in logarithmic form:
(a) 24  16
(b) 103  1000
(c) 93/2  27
(d) 251/2 
1
5
(e) e 2  7.34
(f) e2.0258  10
4. Write the following in index form:
(a) log 2 8  3
(b) log 9 3 
1
2
(c) log 0.1  1
(d) log16 8 
3
4
(e) ln 6  1.792
(f) ln12.18  2.5
5. Without the use of a calculator evaluate the following :
(a) log 2 16
(b) log5 125
(c) log 9 3
(d) log 0.001
(e) log 2
1
8
◙ EP .Program – Strisuksa School Roi-et.
Mathematics
6. Exponential and Logarithmic Functions. page 5
(f) log a a
(g) log a a 2
1
a
6. Use a calculator to find the following :
(a) log 2 3
(h) log a
(b) log 4 9
(c) log 5 2
(d) log3 0.002
7. Simplify the following expression by writing them as single logarithms:
(a) log x  log 2
(b) log 2 x  log 2 3x
(c) log 2 x  3log 2 x
(d) log x  2 log 3x
(e) log x  1
(f) log 2 x  2
8. Solve the following equations:
(a) 2 x  3
(b) 10 x  6
(c) 3x1  4
(d) 22 x1  413x
◙ EP .Program – Strisuksa School Roi-et.
Mathematics
6. Exponential and Logarithmic Functions. page 6
(e) 4 x  5  2 x  6  0
(f) 3x  5x1
(g) 2 x  3x1  10
(h) 2 x  5  3x
(i) e2 x  7e x  10  0
(j) e x  e x  4
9. Write the following equation so that y is the subject.
(a) log x  log y  3
(b) log y x  3
10. Write the following equations:
(a) log 2 x  7
(b) log 2 x  log 2 ( x  1)  log 2 6
(c) log x  log( x  2)  2
(d) 2log3 x  5
(e) log3  log9 x  2
(f) log x 4  5
◙ EP .Program – Strisuksa School Roi-et.
Mathematics
6. Exponential and Logarithmic Functions. page 7
(g) ln x  3
(h) ln x  ln x  3
11. A population is growing at 5% each year. How long will it be before the population
has tripled?
12. A man invests 1, 200 in a building society at 9% compound interest. When will he
have 2, 000 ?
13. A radio-active isotope decays so that each year it loses
1
of its mass. What is its
5
half-life? (The time when it has list haft its mass?)
14. The haft-life of an isotope is 8 years. How much is lost after 1 year? How long does
9
9
it take to lose
of it take to lose
of its mass?
10
10
6.2 Examination Questions
1.
(i) Write the cube root of 4 as a power of 2 .
Hence evaluate
your
 
3
4
4.5
without using a calculator or tables, Showing the steps of
working.
2
 x3 y 
x4 y 2
(ii) Simplify 
 
z
 z 
◙ EP .Program – Strisuksa School Roi-et.
Mathematics
6. Exponential and Logarithmic Functions. page 8
(iii) Simplify logb (ab2 )  logb a .
2.
(i)
Express as a single logarithm in its simplest form
log10 5  3log10 25  2log10 50 .
(ii)
Solve the equation 3  22 x  7  2 x  2  0 .
3.
(a) Express as a single logarithm in its simplest form log10 5  3log10 25  2log10 50 .
1
x
(b) If x x  30 , without attempting to evaluate x find the values of x 2 and x 3 x .
(c) Solve the equation (17.1) x  9
4.
(a) Solve the equation: log3 x  1  log 3 (18  x)
(b) Given that 3  a k , express in terms of k ,
3
(i) log a
a
(ii) log 9 3a
5.
Solve each of the equations, to find x in terms of a , where a  0 and a  e2 x 1 .
(i) a x  e 2 x 1
(ii) 2ln(2 x)  1  ln a .
6.
Evaluate in terms of ln 2
(i) ln 2  ln(2)2   ln(2n ) 

(ii)
  ln 2 
n 1
n
 ln(2100 )
◙ EP .Program – Strisuksa School Roi-et.
Mathematics
6. Exponential and Logarithmic Functions. page 9
Common errors
1. Indices
Algebraic and arithmetic mistakes are very common when dealing with indices. Pay
regard to the following :
a n  a m  a mn
a
a1/n 
n
n
a  a n
2. Logarithms
Similar mistakes arise when dealing with logarithms. Pay regard to the following:
log a  log b  log(a  b)
log a  log b  log ab
3. Compound Interest
In problem involving compound interest , or percentage growth or decay, remember
that the original number id multiplied by the same amount each tear . Do not add a
constant amount .
Solution (to exercise 6.1.2)
1.
(a)
(b)
(c)
(d)
4
1000
16
16
2.
(a) x 4
(b) y 3
(c) 59 x
(d) 220 n
3.
(a) 4  log 2 16
(b) 3  log10 1000
3
 log9 27
2
1
1
(d)   log 25
2
5
(e) ln 7.34  2
(f) ln10  2.0258
(c)
4.
(a) 23  8
◙ EP .Program – Strisuksa School Roi-et.
(b) 91/2  3
(c) 101  0.1
(d) 163/4  8
(e) e1.792  6
(f) e2.5  12.18
5.
(a) 4
(b) 3
1
(c)
2
(d) 3
(e) 3
(f) 1
(g) 2
1
(h) 
2
6.
(a)
(b)
(c)
(d)
1.585
1.585
0.431
5.657
7.
(a) log 2x
1
3
(c) log 2 x4
(b) log 2
(d) log 9x 2
(e) log10x
(f) log 2 4x
8.
(a) 1.585
(b) 0.778
(c) 0.262
3
(d)
8
(e) 1 or 1.585
(f) 3.15
(g) 0.672
(h) 0.898
(i) 1.609 or 0.693
Mathematics
6. Exponential and Logarithmic Functions. page 10
◙ EP .Program – Strisuksa School Roi-et.
Mathematics
6. Exponential and Logarithmic Functions. page 11
(j) 1.317
9.
(a) y 
1000
x
(b) y  3 x
10.
(a) 128
(b) 2
200
(c)
99
(d) 15.59
(e) 81
(f) 1.3195
(g) 20.09
(h) 0.135
11. 22.5 years
12. 5.93 years
13. 3.1 year
14. 0.083, 26.6 years
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References:
Solomon, R.C. (1997), A Level: Mathematics (4th Edition) , Great Britain, Hillman
Printers(Frome) Ltd.
More: (Thai exam.)
http://home.kku.ac.th/wattou/service/m456/06.pdf