# Applications of logarithms

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Elementary FunctionsPart 3, Exponential Functions & Logarithms

Lecture 3.6a, Applications of Logarithms

Dr. Ken W. Smith

Sam Houston State University

2013

Smith (SHSU) Elementary Functions 2013 1 / 21

Using logarithms to solve Newtons Law of Cooling

Recall Newtons Law of Cooling in which the ratio of differences intemperature decays exponentially:

T (t)TaT0Ta = e

kt

Here k is a constant that depends on the liquid and the environment. Wesolve for the temperature T at time t.

T (t) = Ta + (T0 Ta)ekt

(Here we use the notation of the textbook by Stitz and Zeager.)In most applications of this law, we need to solve for k and/or t. Sinceboth k and t are exponents, we must use logarithms.

Smith (SHSU) Elementary Functions 2013 2 / 21

Applications of logarithms

A worked example.Suppose the water in a hot tub is heated to 150. After the heater isturned off, the hot tub takes an hour to cool to 120. The temperature ofthe surrounding air is 80. Use Newtons Law of Cooling:

T (t) = Ta + (T0 Ta)ekt

to find the temperature of the hot tub after 3 hours.

Solution. At time t = 1 we have

120 = 80 + (150 80)ek

Subtracting 80 from both sides and then dividing by 70 gives

40 = 70ek

47 = e

k.

Taking the natural log of both sides we have

ln(47) = k

k = ln(4/7) = ln(7/4) .Smith (SHSU) Elementary Functions 2013 3 / 21

Applications of logarithms

So the equation for T at time t is

T (t) = 80 + (150 80)et ln(4/7).

We may simplify this by replacing (150 80) by 70. We can also (if wewish) replace et ln(4/7) by (eln(

47))t = (47)

t so that

T (t) = 80 + 70et ln(4/7) = 80 + 70(47)t.

At time t = 3,

T (3) = 80 + 70(4/7)3 80 + 13.0612 = 93.0612

Smith (SHSU) Elementary Functions 2013 4 / 21

Applications of logarithms

Now let us suppose that we want to know the time t this hot tub will cooldown to 85.

Solution.Here we need to solve for t in the equation

85 = 80 + 70(47)t.

Subtract 80 from both sides and divide by 70 and simplify 570 =114 so that

114 = (4/7)

t

Taking the natural log of both sides we have

ln( 114) = ln((47)

t).

Rewriting the lefthand side as ln( 114) = ln 1 ln 14 = ln 14 andrewriting the righthand side as ln(47

t) = t ln 47 gives us

ln 14 = t ln 47 .

So t = ln 14ln 4

7

= ln 14ln 7

4

4.7158 hours.Smith (SHSU) Elementary Functions 2013 5 / 21

Radioactive half-life

A radioactive isotope decays at random, but with a probability computableover an interval of time. The proportion of radioactive material thatdecays in a particular interval depends on the isotope. Since the amountof material that decays is proportional to the total amount of material,radioactive decay is an example of exponential decay.Here is an example. A tree that died in the forest 83 years ago is examinedfor carbon 14. We assume that the ratio amount of carbon 14 in the treewhen it died is compatible with that in the atmosphere at the time butnow, 83 years later, we discover that one percent of the carbon 14 hasdecayed to nitrogen 14. We can model this by an exponential decayequation:

P (t) = P0ekt

where P (t) represents the mass of carbon 14 at time t and P0 representsthe mass of carbon 14 at time t = 0. (Here we assume that k is positive,so that the exponential function is decaying, not growing.)In this case, with t = 83, we have that P (t) = 0.99P0. So our equation is

0.99P0 = P0e83k.Smith (SHSU) Elementary Functions 2013 6 / 21

Radioactive half-life

We discovered that

0.99P0 = P0e83k.

We may divide both sides of this equation by P0 to obtain

0.99 = e83k

and then take the natural log of both sides

ln(0.99) = 83k.

Therefore k = ln 0.9983 0.000121So the amount of carbon 14 t years after the death of the specimen can beapproximated by

P (t) = P0e0.000121t.

Smith (SHSU) Elementary Functions 2013 7 / 21

Applications of logarithms

Sample Problem. Suppose the amount of carbon 14 in a specimen canbe measured by the equation above. From the equation, what is thehalf-life of carbon 14?

Solution.In this case, we solve for t in the equation

12P0 = P0e

0.000121t.

Dividing both sides by P0 we have12 = e

0.000121t.

Taking the natural log of both sides we have

ln(12) = 0.000121t,or (using the power rule of logs)

ln(2) = 0.000121t,

t = ln(2)0.000121 5728.5years.

(The correct half-life for carbon 14 (according to Wikipedia is 5730 40years, so this answer is compatible with our understanding of carbon 14.)

Smith (SHSU) Elementary Functions 2013 8 / 21

http://en.wikipedia.org/wiki/Carbon-14

Applications of logarithms

In the next presentation we will continue with applications of logarithms todoubling times of investments and similar problems where we are, onceagain, solving for an exponent.

(END)

Smith (SHSU) Elementary Functions 2013 9 / 21

Elementary FunctionsPart 3, Exponential Functions & Logarithms

Lecture 3.6b, Applications of Logarithms: Doubling Time

Dr. Ken W. Smith

Sam Houston State University

2013

Smith (SHSU) Elementary Functions 2013 10 / 21

Doubling time for an investment

Just as radioactive isotopes decay exponentially, investment of moneytypically grows in proportion to the money invested and so financialinvestments are an example of exponential growth.In financial investments, we are often interested in the doubling time ofan investment.It this case we solve for time where the future value of the investment is2P0 where P0 is the amount of our initial investment.Here are some sample problems.

Smith (SHSU) Elementary Functions 2013 11 / 21

Doubling time for an investment

Worked problems on doubling time.Find the doubling time for the following investments.

1 An annual interest rate of 9%, compounded monthly.Solution. Here the monthly interest rate is r = 0.09/12 = 0.0075and time n is measured in months, the length of the interest period.We solve for the exponent n in the equation

2P0 = P0(1.0075)n,

or, after dividing by P0,

2 = (1.0075)n.

Taking the natural log of both sides we have that

ln 2 = ln((1.0075)n).

Using properties of logs, we may rewrite this as

ln 2 = n ln(1.0075)

n = ln 2ln 1.0075 92.766.Here time is measured in months so the doubling time is almost93 months or 7 years and 9 months .Smith (SHSU) Elementary Functions 2013 12 / 21

Doubling time for an investment

2 An annual interest rate of 24%, compounded monthly.Solution. We solve for n in the equation

2P0 = P0(1.02)n.

(Here r = 0.24/12 = 0.02 and time n is measured in months, thelength of the interest period.) Dividing by P0 and taking the naturallog of both sides we have that

ln 2 = ln((1.02)n).

Using properties of logs, we may rewrite this as

ln 2 = n ln(1.02).

and so

n = ln 2ln 1.02 35.0028.Since time here is measured in months then the doubling time is35 months or 2 years and 11 months .

Smith (SHSU) Elementary Functions 2013 13 / 21

Doubling time for an investment

3 An annual interest rate of 9%, compounded continuously.Solution. We solve for t in the equation 2 = e0.09t where t is timemeasured in years. Taking the natural log of both sides we have thatln 2 = 0.09t and so t = ln 20.09 =

100 ln 29 which is about 7.702 years .

Notice that this answer is close to the answer in problem 1.

Smith (SHSU) Elementary Functions 2013 14 / 21

Doubling time for an investment

4 An annual interest rate of 24%, compounded continuously.Solution. We solve for t in the equation 2 = e0.24t. Taking thenatural log of both sides we have that ln 2 = 0.06t and sot = ln 20.24 =

100 ln 224 which is about 2.88 years . Notice that this answer

is close to the answer in problem 2.

Smith (SHSU) Elementary Functions 2013 15 / 21

The Rule of 72

The Rule of 72In general, the doubling time for an investment solves for t in theexponential equation

2 = (1 + r)t.

If we apply the natural log to this equation, we find that

t = ln 2ln(1+r) .

For interest rates close to 8%, this formula can be accuratelyapproximated by the formula

f(t) =72

100r.

In other words, the doubling time is about what one would get by takingthe interest rate, in percents, an dividing it into 72. Since it is fairly easyto divide numbers into 72, this rough rule of thumbs shows up in businessdiscussions as the rule of 72.

For example, when does a 4% interest rate double? Since 724 = 18, a 4%rate should double in about 18 years.

Smith (SHSU) Elementary Functions 2013 16 / 21

Applications of logarithms

Use the Rule of 72 to approximate the following:

1 The doubling time of a 3% investment,

2 The doubling time of an 8% investment,

3 The doubling time of a 9% investment,

4 The doubling time of a 24% investment,

Solutions.

1 Using the Rule of 72 we estimate that a 3% investment should doublein approximately 723 = 24 years. (The exact answer is slightly morethan 23 years.)

2 An 8% investment should double in approximately 728 = 9 years.

3 Using the Rule of 72 we estimate that an 9% investment shoulddouble in approximately 729 = 8 years. (The exact answer is slightlyunder 8.)

4 Using the Rule of 72 we estimate that a 24% investment shoulddouble in approximately 7224 = 3 years. (Our exact calculations abovegave just under 3 years. )

Smith (SHSU) Elementary Functions 2013 17 / 21

The logistic function

Suppose we believ

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