PreCalculusName: ______

LOGARITHMS #4Period: _____

Worksheet Lg4

Applications of Logs and Exponentials

I.Benford’s Law

In 1993, in State of Arizonav. Wayne James Nelson, the accused was found guilty of trying to defraud the state of nearly $2 million. Nelson, a manager in the office of the Arizona State Treasurer, argued that he had diverted funds to a bogus vendor to demonstrate the absence of safeguards in a new computer system. The amounts of the 23 checks issued seemed to have been chosen to given the appearance of randomness: None of the check amounts were duplicated; there were no round numbers; and all the amounts included cents. However, subconsciously, the manager repeated some digits and digit combinations, and there was a tendency towards higher digits. An analysis of this data showed that it was extremely unlikely to have come from real sources. In particular, an application of what is known as Benford’s Law, indicated that the set of leading digits were extremely anomalous.

So what is this Benford’s Law? Well, in its most basic form, Benford’s law states that the distribution of leading digits in any collection of real data involving large numbers will follow a logarithmic law. Specifically, the probability of finding a leading digit d (from 1 to 9) will equal log (d + 1) – log d. For example, the probability that the leading digit of the population of a city chosen at random from an almanac will be 3 is equal to log 4 – log 3 = 0.125, and the probability that it will be 7 is equal to log 8 – log 7 = 0.058. This law is used by the IRS and many other agencies to detect fraud, since most people think the digits should appear more or less uniformly with equal probability 1/9.

More precisely, Benford’s law applies when the distribution of the data rises for a while and then falls off slowly. It would look something like this (think of the graph as a “smoothed out histogram”):

The reason it works this way can be explained as follows. If d is the leading digit of some big number x, then x must be between and , for some whole number n. For example, if x = 32,150, then the leading digit is 3, and x is between and . If we take the base-ten logs, we have , which simplifies to

, which in turn is equivalent to .

Now is a number between 0 and 1, and we would expect it to be uniformly distributed on that interval. But the intervals determined by and partition the interval from 0 to 1 according to the following distribution:

Problems for you to solve:

1.Sally and Al are looking through some census data on family incomes. Sally says more than 30% of the data will have a leading digit of 1. Is Sally right? Why or why not?

2.What is the probability that the leading digit of the population of a randomly picked county in the United States will be 7?

3.What is the probability that the leading digit of some census data is greater than 5?

4.A family is chosen at random from the population of the United States, and this family’s income is determined. The probability that the leading digit is 2 is exactly 1/2 the probability that the

digit belongs to which of the following sets?

A. { 2, 3 }B. { 3, 4 }C. { 4, 5, 6, 7, 8 }D. { 5, 6, 7, 8, 9 }E. { 4, 5, 6, 7, 8, 9 }

5.Some references give the following formula for the probability that the leading digit of some data drawn from the census will be d: , where .

Verify that this formula is the same as .

II.Decibels

Sound waves are produced by the vibration of an object. For example, a vibrating guitar string forces surrounding air molecules to be compressed and expanded, creating a pressure disturbance consisting of an alternating pattern of compressions and rarefications. Energy is carried by this disturbance and enters our ears. The intensity of this energy is measured in watts per square meter. This is called the intensity of the sound. A sound with an intensity of watts per square meter corresponds to a sound which will displace particles of air by a mere one-billionth of a centimeter, and amazingly the human ear can detect such a sound. This is the threshold of hearing. The most intense sound which the ear can safely detect without suffering any physical damage is more than one billion times as intense as the threshold of hearing.

Since the range of intensities which the human ear can detect is so large, the scale which is frequently used to measure intensity is the decibel scale. A decibel is 10 times the base-ten log of

the ratio of the intensity of a sound to the threshold of hearing: , where x is the intensity of the sound, and is the threshold of hearing. The following table lists some

common sounds with an estimate of their intensity and decibel level:

(reference:

Source / Intensity (watts per m2) / Decibels
Threshold of Hearing / 10–12 / 0
Rustling Leaves / 10–11 / 10
Whisper / 10–10 / 20
Normal Conversation / 10–6 / 60
Busy Street Traffic / 10–5 / 70
Vacuum Cleaner / 10–4 / 80
Large Orchestra / 10–3 / 98
Walkman at Maximum Volume / 10–2 / 100

More problems for you to solve:

6.The buzz of a mosquito produces an intensity of 40 decibels. How many mosquitoes would match the intensity of normal conversation (60 dB)? Hint: A single mosquito produces x watts per square meter, where . Two mosquitoes produce 2x watts per square meter, and so on. A human voice in normal conversation produces y watts per square meter, where . How many x’s are needed to equal one y?

7.The threshold of pain is around 130 decibels. How many watts per square meter is this?

8.On a good night, the front row of a certain rock concert has a sound level of 120 dB. A walkman produces 100 dB at maximum volume. How many Walkmen would be needed to produce the same intensity as the front row at this concert?

III.Interest

When you deposit money in a savings account that compounds interest periodically for N periods a year (N = 12 if monthly, 52 if weekly, etc.), and if k is the periodic interest rate as a decimal (for example, if compounded monthly and r is the APR as a decimal), then the balance at the end of n compounding periods is given by , where A is the amount you deposited. This can also be written as , where . If the interest is compounded continuously (like every second), then the formula is , where r is the APR as a decimal and y is the number of years. Now there is usually some confusion about the term “annual percentage rate.” This is not the rate at which the interest accumulates each year unless it is compounded annually. Instead, it is a standard for reporting rates, and the true rate of increase depends on the number of compounding periods. For example, if the APR is 12% and interest is compounded monthly, then at the end of one year a deposit of A = $100 will have grown to , so the actual interest was $12.68, and the true percentage increase was 12.68%.

More problems for you to solve:

9.If a savings account has an APR of 12% and compounds interest weekly, what is the true annual interest rate?

10.If a savings account has an APR of 12% and compounds interest continuously, what is the true annual interest rate?

11.If you deposit $2000 in a savings account with an APR of 4%, compounded monthly, how long will it take for the balance to reach $3000?

12.If you deposit $2000 in a savings account with an APR of 4%, compounded continuously, how long will it take for the balance to reach $3000?

13.If you deposit money in a money market with an APR of 5%, compounded monthly, how long must you leave it there for the amount to double?

14.If you deposit money in a money market with an APR of 5%, compounded continuously, how long must you leave it there for the amount to double?

IV.The Logistic Model (See pages 517-519 of your book.)

This is a mathematical model often used to describe growth of populations. It is similar to the exponential model, but takes account of the fact that real populations have a limit to their sizes. This limit is called the “carrying capacity” of the population. For example, in the development of an embryo, a fertilized ovum splits and the cell count grows: 1, 2, 4, 16, 32, and so on. This is exponential growth. But the fetus can grow only as large as the uterus can hold; thus other factors start slowing down the increase in the cell count, and the rate of growth slows (but the baby is still growing, of course). After a suitable time, the child is born and keeps growing. Ultimately, the cell count is stable; the person's height is constant; the growth has stopped, at maturity.

The mathematical equation for the logistic model is , where c is the carrying capacity, r is the proportional rate of growth, x is the time factor, y is the size of the population at time x, and a is related to c and the size y0 of the population at time x = 0 by .

Example:Suppose a population that initially has 1,000 members grows according to the logistic model with carrying capacity 6,000 and r = 0.05 per year (that is, the population initially increases at a rate of 5% of its size each year). Then the equation that gives its size after x years is

More problems for you to solve:

15.Let. Find the value of y when:

a. x = 0b. x = 20c. x = 40

d. x = 100e. x = 1,000f.

16.Let. Find the value of x when:

a. y = 1000b. y = 2000

c. y = 3000d. y = 5000

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