a)Very Basic Problems related to Logarithim
1)Use your calculator to find:
a) Answer=-4
log 6 / / 11296
/
b) Answer=-2
log 5 / / 125
/ / =
c) Answer=(3/2)
log 4 / 8 / =d) Answer=4
log (1/4) / / 1256
/
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2) Solve for unknown
a)
log (x + 188) = 2.3483b)
log (12.53 × 10d) = 67.098c)
log (x - 41) = 2.2718d)
2x - 3 / = / 3x - 3.3691e)
3x + 0.3347 / = / 5x - 0.4063Problem 3
The population P of a city increases according to the formula
P = 5000 e a t
where t is in years and t = 0 corresponds to 1980. In 1990, the population was 10000. Find the value of the constant a and approximate your answer to 3 decimal places.
If you were to draw the population vs time what kind paper would you use?
Problem 4
The populations P1 and P2 of two cities are given by the formulae
P1 = 10000 e k t
P2 = 20000 e 0.01 t
where k is a constant and t is the time in years with t = 0 corresponding to the year 2000. Find constant k so that the two populations are equal in the year 2020 and approximate your answer to 3 decimal places.
Problem 5
The level of sound D in decibels is defined as folows
D = 10 log( I / 10 -16 )
where I is the sound intensity in watts per centimeters squared. Determine the level in decibels of a sound with intensity I = 10 -8 watts/cm 2.
Problem 6
Two sounds of intensities I1 and I2 have decibel levels of 60 and 80 respectively. Use the formula for decibel level given in problem 5 to determine the ratio of the intensities I2 / I1?
Problem 7
The spread of a virus through a city is modeled by the function
N = 15000 / [ 1 + 100 e -0.5 t ]
where N is the number of people infected by the virus after t days. How many days it takes for 2000 people of this city to be infected with the virus? (approximate your answer to 3 decimal places).
Problem 8
The amount of a radioactive material decays according to the formula
A(t) = A o e -k t
where A o is the initial amount, k is a positive constant and t is the time in days. Find a formula for the half life of the material.
Problem 9
A bacteria population doubles every eight minutes. If the population begins with one cell, how long will it take to grow to 512 cells?
Problem 10
A bacteria population doubles every eight minutes. If the population begins with one cell, how long will it take to grow to 2,097,152 cells?
Problem 11
A translucent plastic paper reduces the intensity of light that passes through it by eight percent. Amy wants to combine a number of these papers to only allow at the most forty percent of light to pass through. How many translucent plastic papers should Amy use?
Problem 12
Brad spent $1,200 on a credit card this month. The credit card charges 24.77% interest compounded continuously. If Brad does not have to pay any monthly minimums and makes no payments, how long will it take until Brad owes the credit card company $2,400?
Problem 13
Jill's investment in a bank account will be worth $3,746.42 in six months. The current value of Jill's account is $3,600. If the bank pays a fixed rate that is compounded monthly, how much will the account be worth in two years?
Problem 14
The half-life of a radioactive substance is one hundred fifty-three days. How many days will it take for seventy percent of the substance to decay?
For Problems 15-19 assume:
A = Pekt
Problem 15
A biologist is researching a newly-discovered species of bacteria. At time t = 0 hours, he puts one hundred bacteria into what he has determined to be a favorable growth medium. Six hours later, he measures 450 bacteria. Assuming exponential growth,
A = Pekt
what is the growth constant "k" for the bacteria? (Round k to two decimal places.)
Problem 16
A certain type of bacteria, given a favorable growth medium, doubles in population every 6.5 hours. Given that there were approximately 100 bacteria to start with, how many bacteria will there be in a day and a half?Assuming exponential growth,
A = Pekt
Problem 17
Radio-isotopes of different elements have different half-lives. Magnesium-27 has a half-life of 9.45 minutes. What is the decay constant for Magnesium-27? Round to five decimal places.
Problem 18
Some people are frightened of certain medical tests because the tests involve the injection of radioactive materials. A hepatobiliary scan of my gallbladderinvolved an injection of 0.5 cc's (or about one-tenth of a teaspoon) of Technetium-99m, which has a half-life of almost exactly 6 hours. While undergoing the test, I heard the technician telling somebody on the phone that "in twenty-four hours, you'll be down to background radiation levels." Figure out just how much radioactive material remained from mygallbladderscanafter twenty-four hours.
Problem 19
Carbon-14 has a half-life of 5730 years. You are presented with a document which purports to contain the recollections of a Mycenaean soldier during the Trojan War. The city of Troy was finally destroyed in about 1250 BC, or about 3250 years ago. Carbon-dating evaluates the ratio of radioactive carbon-14 to stable carbon-12. Given the amount of carbon-12 contained a measured sample cut from the document, there would have been about 1.3 × 10–12 grams of carbon-14 in the sample when the parchment was new, assuming the proposed age is correct. According to your equipment, there remains 1.0×10–12 grams. Is there a possibility that this is a genuine document? Or is this instead a recent forgery?