4.4 the logarithm functions t

The Logarithmic Functions
Example B. Rewrite the log-form into the exp-form.
a. log3(1/9) = –2  3-2 = 1/9
b. 2w = logv(a – b)  v2w = a – b
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16  log4(16) = 2
b. w = u2+v  logu(w) = 2+v
To convert the exp-form to the log–form:
b = y
x
logb( y ) = x→
To convert the log–form to the exp–form:
b = y
x
logb( y ) = x→
The output of logb(x), i.e. the exponent in the defined
relation, may be positive or negative.

Example C.
a. Rewrite the exp-form into the log-form.
4–3 = 1/64
8–2 = 1/64
log4(1/64) = –3
log8(1/64) = –2
exp–form log–form
b. Rewrite the log-form into the exp-form.
(1/2)–2 = 4log1/2(4) = –2
log1/2(8) = –3
exp–formlog–form
(1/2)–3 = 8
Example F. Solve for x
a. log9(x) = –1
Drop the log and get x = 9–1.
So x = 1/9

b. logx(9) = –2
Drop the log and get 9 = x–2, i.e. 9 =
So 9x2 = 1
x2 = 1/9
x = 1/3 or x= –1/3
Since the base b > 0, so x = 1/3 is the only solution.
The Common Log and the Natural Log

(1, 0)
(2, 1)
(4, 2)
(8, 3)
(16, 4)
(1/2, -1)
(1/4, -2)
y=log2(x)
Graphs of the Logarithmic Functions
1/4 -2
1/2 -1
1 0
2 1
4 2
8 3
x y=log2(x)
Recall that the domain of logb(x) is the set of all x > 0.
Hence to make a table to plot the graph of y = log2(x),
we only select positive x’s. In particular we select x’s
related to base 2 for easy computation of the y’s.
x
y

x
y
(1, 0)
(8, -3)
To graph log with base b = ½, we have
log1/2(4) = –2, log1/2(8) = –3, log1/2(16) = –4
(4, -2)
(16, -4)
y = log1/2(x)
x x
y
(1, 0)(1, 0)
y = logb(x), b > 1
y = logb(x), 1 > b
Here are the general shapes of log–functions.
y
(b, 1)
(b, 1)

3x2
y
log( ) = log( ), by the quotient rule
= log (3x2) – log(y1/2)
product rule power rule
= log(3) + log(x2) – ½ log(y)
= log(3) + 2log(x) – ½ log(y)
3x2
y
3x2
y1/2
Properties of Logarithm
a. Write log( ) in terms of log(x) and log(y).
log(3) + 2log(x) – ½ log(y) power rule
= log(3) + log(x2) – log(y1/2) product rule
= log (3x2) – log(y1/2)= log( )3x2
y1/2
b. Combine log(3) + 2log(x) – ½ log(y) into one log.
Example G.
quotient rule

we have that:
a. logb(expb(x)) = x or logb(bx) = x
b. expb(logb(x)) = x or blog (x) = x
b
Example H. Simplify
a. log2(2-5) = -5
b. 8log (xy) = xy
c. e2+ln(7) = e2·eln(7) = 7e2
8

1.
Exercise A. Rewrite the following exp-form into the log-form.
2. 3.
4. 5. 6.
7. 8. 9.
10.
Exercise B. Rewrite the following log–form into the exp-form.
52 = 25 33 = 27
1/25 = 5–2 x3 = y
y3 = x ep = a + b
e(a + b) = p 10x–y = z11. 12.
1/25 = 5–2
1/27 = 3–3
1/a = b–2
A = e–rt
log3(1/9) = –2 –2 = log4(1/16)13. 14. log1/3(9) = –215.
2w = logv(a – b)17. logv(2w) = a – b18.log1/4(16) = –216.
log (1/100) = –2 1/2 = log(√10)19. 20. ln(1/e2) = –221.
rt = ln(ert)23. ln(1/√e) = –1/224.log (A/B) = 322.

Exercise C. Convert the following into the exponential form
then solve for x.
logx(9) = 2 x = log2(8)1. 2. log3(x) = 23.
5. 6.4.
7. 9.8.
logx(x) = 2 2 = log2(x) logx(x + 2) = 2
log1/2(4) = x 4 = log1/2(x) logx(4) = 1/2
11. 12.10.
13. 15.14.
ln(x) = 2 2 = log(x) log(4x + 15) = 2
In(x) = –1/2 a = In(2x – 3) log(x2 – 15x) = 2

Ex. D
Disassemble the following log expressions in terms of
sums and differences logs as much as possible.
5. log2(8/x4) 6. log (√10xy)
y√z3
2. log (x2y3z4)
4. log ( )
x2
1. log (xyz)
7. log (10(x + y)2) 8. ln ( )
√t
e2
9. ln ( )
√e
t2
10. log (x2 – xy) 11. log (x2 – y2) 12. ln (ex+y)
13. log (1/10y) 14. log ( )x2 – y2
x2 + y2
15. log (√100y2)
3
3. log ( )
z4
x2y3
16. ln( )x2 – 4
√(x + 3)(x + 1)
17. ln( )(x2 + 4)2/3
(x + 3)–2/3(x + 1) –3/4

E. Assemble the following expressions into one log.
2. log(x) – log(y) + log(z) – log(w)
3. –log(x) + 2log(y) – 3log(z) + 4log(w)
4. –1/2 log(x) –1/3 log(y) + 1/4 log(z) – 1/5 log(w)
1. log(x) + log(y) + log(z) + log(w)
6. –1/2 log(x – 3y) – 1/4 log(z + 5w)
5. log(x + y) + log(z + w)
7. ½ ln(x) – ln(y) + ln(x + y)
8. – ln(x) + 2 ln(y) + ½ ln(x – y)
9. 1 – ln(x) + 2 ln(y)
10. ½ – 2ln(x) + 1/3 ln(y) – ln(x + y)

Continuous Compound Interest
F. Given the following projection of the world
populations, find the growth rate between each
two consecutive data.
Is there a trend in the growth rates used?

(Answers to the odd problems) Exercise A.
Exercise B.
13. 3−2 = 1/9 15.
1
3
−2
= 9 17. 𝑦2𝑤
= 𝑎 − 𝑏
19. 10−2 = 1/100 21. 𝑒−2 = 1/𝑒2 23. 𝑒 𝑟𝑡 = 𝑒 𝑟𝑡
1. 𝑙𝑜𝑔5 25 = 2 3. 𝑙𝑜𝑔3 27 = 3
7. 𝑙𝑜𝑔 𝑦(𝑥) = 3 9. 𝑙𝑜𝑔 𝑒 𝑎 + 𝑏 = 𝑝
5. 𝑙𝑜𝑔391/27) = −3
11. 𝑙𝑜𝑔 𝑒 𝐴 = −𝑟𝑡
Exercise C.
1. 𝑥 = 3 3. 𝑥 = 9 5. 𝑥 = 4 7. 𝑥 = −2
9. 𝑥 = 16 11. 𝑥 = 100 13. 𝑥 =
1
𝑒
15. 𝑥 = −5, 𝑥 = 20

Exercise D.
5. 𝑙𝑜𝑔2 8 − 4𝑙𝑜𝑔2(𝑥)
1. log(𝑥) + log(𝑦) + log(𝑧)
7. log(10) + 2log(𝑥 + 𝑦)
9. 2ln(𝑡) − 1/2ln(𝑒) 11. log(𝑥2 – 𝑦2)
13. log(1) − 𝑦𝑙𝑜𝑔(10)
3. 2log(𝑥) + 3log(𝑦) − 4log(𝑧)
15. 1/3log(100) + 2/3log(𝑦)
17. 2/3ln(𝑥 + 4) + 2/3ln(𝑥 + 3) + 3/4ln(𝑥 + 1)
Exercise E.
3. log
𝑦2 𝑤4
𝑥𝑧3
1. log(𝑥𝑦𝑧𝑤) 5. 𝑙𝑜𝑔 (𝑥 + 𝑦)(𝑧 + 𝑤)
7. 𝑙𝑛
𝑥(𝑥+𝑦)
𝑦
9. 𝑙𝑛
𝑒𝑦2
𝑥

4.4 the logarithm functions t

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