10.5 Problems
Write the expression in logarithmic form
1. 53 = 125
5. 3−2 =
1
9
Write in exponential form
7. log 2 32 = 5
1
9.ln ( ) = −1
𝑒
11. log 100,000 = 5
Evaluate without using a calculator
13. log 8 64
15. log 4 64
17. log 2
1
16
3
19. log 2 √
1
4
21. ln 𝑒
5
23. ln 𝑒 3
25. Is “the logarithm of base 3 of 4” written as log 4 3 or log 3 4?
Use properties of logarithms to write each expression as a sum, product or
difference of logarithms
27. log 5 (3𝑘)
29. log 3
3𝑝
5𝑘
3√5
31. ln⁡( 3 )
√6
Suppose log 𝑏 2 = 𝑎 and log b 3 = 𝑐. Find:
33.log 𝑏 32
35.log 𝑏 (72𝑏)
Use natural logarithms to evaluate each term to the nearest thousandth
37. log 5 30
39. log1.2 0.95
Solve:
41. log 𝑥 36 = −2
43. log 8 16 = 𝑧
45.log 𝑟 5 =
1
2
47.log 5 (9𝑥 − 4) = 1
48.log 4 𝑥 − log 4 (𝑥 + 3) = −1
51.log 3 (𝑥 − 2) + log 3 (𝑥 + 6) = 2
57.⁡⁡⁡𝑒 2𝑦 = 15
59.⁡⁡⁡10𝑒 3𝑥−7 = 100
Find the domain
62. 𝑓(𝑥) = log⁡(5 − 𝑥)
77. When a pharmaceutical drug is injected into the bloodstream, its
concentration at time t can be approximated by 𝐶(𝑡) = 𝐶0 𝑒 −𝑘𝑡 , where 𝐶0
is the concentration at t=0. Suppose the drug is ineffective below a concentration
of 𝐶1 and harmful above a concentration of 𝐶2 . Then it can be shown that the
drug should be given at time intervals of T where
𝑇=
1 𝐶2
ln⁡( )
𝑘 𝐶1
A certain drug is harmful at concentrations five times the concentration below
which it is ineffective. At noon, an injection of the drug results in a concentration
of 2mg per liter of blood. Three hours later, the concentration is down to 1mg
per liter. How often should the drug be given?
79. The U.S. census bureau has reported that the United States is becoming more
diverse. The projected Hispanic population (in millions) was modeled by the
exponential function
ℎ(𝑡) = 37.79(1.021)𝑡
where t corresponds to 2000 and t is in [0,50].
(a) Estimate in what year the Hispanic Population will double the 2005 population
of 42.69 million.
(b)The projected U.S. Asian population ( in millions) was modeled by the
exponential function
ℎ(𝑡) = 11.14(1.023)𝑡
Where t=0 corresponds to 2000 and t is in [0,50]. Estimate in what year the Asian
population will double the 2005 population of 12.69 million.
80. The number of years N(r) since two independently evolving languages split off
from a common ancestral language is approximated by
𝑁(𝑟) = ⁡ −5000⁡ln⁡(𝑟)
Were r is the proportion of words from the ancestral language that are common
to both languages now
Find
a. N(0.9)
b. N(0.5)
c. N(0.3)
d. How many years have elapsed since the split if 70% of the words of the
ancestral language are common to both languages today?
e. if two languages split off from a common ancestral language about 1000 years
ago find r.
82. The loudest of sounds is measured in a unit called a decibel. To do this, a very
faint sound, called a threshold sound, is assigned an intensity 𝐼0 . If a particular
sound has intensity 𝐼, then the decibel rating of this sound is
10 log
Find the decibel ratings of
a. Whisper 115⁡𝐼0
𝐼
𝐼0
b. Busy street, 9,500,000⁡𝐼0
c. Heavy truck 20m away, 1,200,000,000⁡𝐼0
d. Rock music concert, 895,000,000,000⁡𝐼0
e. Jetliner takeoff, 109,000,000,000,000⁡𝐼0
f. The threshold of sound was identified as 𝐼0 = 0.0002⁡microbars. Use this
definition to identify the sound levels in parts c and d in microbars