Name: ____________________ Pre- Calculus 12 Date: _____________

Chapter 8 – Logarithmic Functions

8.5 – Natural Logarithm and the Number e

Natural Logarithm: The natural logarithm loge(x) is computed using the LN button on your calculator. Its

inverse function, ex, is computed using the same button in conjunction with the 2ND button. The natural

logarithm turns out to be the most convenient one to use in mathematics, because a lot of formulas,

especially in calculus, are much simpler when the natural logarithm is used. The natural logarithm is

abbreviated by ln(x).

The number e, used in science and math, is an irrational number whose value is approximately 2.718, is the

base for the Natural Logarithms, which is abbreviated ln.

You can take antilogarithms of natural logarithms as well. The symbol for the antilogarithm of x is

lnanti x .

Use your calculator to find each of the following.

1. ln 732 2. ln1685 3. ln0.0824

4. ln1.3475anti 5. ln0.0813anti 6. ln3.111anti

Earlier in this chapter when we were describing population growth or radioactive decay, we used phrases

such as “the number of bacteria doubles everyday” or “the substance has a half life of 3 minutes”. These

descriptions are not very precise as the number of bacteria wouldn’t be doubled all of a sudden from the

original amount after a day, and the radioactive substance wouldn’t have half of its mass evaporated in a

click after 3 minutes. These processes happen CONTINUOUSLY. The formulae we used are not precise

enough.

The number e is used in situations when growth (or decay) happens continuously, such as the melting of an

iceberg or population growth for human beings.

The formula for continuous growth or decay is:

ktA P e where A = final amount P = present amount

e = 2.718281828459… …

k = rate (growth / decay) t = time duration

Example 1: The current Canadian population is 33 million. Assume the population is growing

continuously at an average rate of 1.8% per annum.

a) Determine the population, in millions, 20 years from now.

b) How long would it take for the population to double?

To solve equations with base e, scientists turn to natural log (ln). It is logarithm with base e.

xxe lnlog where 0x

The natural logarithm of a number x is written as ln x (pronounced as “lawn x”).

Example 2: A certain radioactive substance disintegrates over time. Starting with 30 grams, the

formula tkeA 30 is used to calculate its mass after t years. Determine k if the substance is to

reduce to one-third of its mass in 25 years.

Example 3: How long would it take for a $1000 deposit to grow to $1 million at 8% compounded

continuously?

Example 4: Using ln on the calculator, evaluate the following natural logs to 5 decimal places.

a) ln 10 b) ln (-3) c) ln (0) d) ln e

Note that the rules for logarithm apply to “ln”.

Example 5: Evaluate the following without a calculator.

a) ln e3

b) ln 1 c) ln 0 d) 2ln xe

Practices: # 1–6 Monetary Problems

# 7– 12 Growth & Decay

1. Find the amount that results from each investment.

a) $240 invested at 12% compounded continuously after 3 yrs.

b) $90 invested at 7% compounded continuously after 3 ¾ yrs.

2. Find the principal needed now to get each amount.

a) To get $1000 after 1 year at 12% compounded continuously.

b) To get $800 after 2 ½ years, 8% compounded continuously.

3. Jerry will be buying a new car for $15000 in 3 years. If an

investment tool is available in the market at a rate of 5%

compounded continuously, how much money should he invest for

now so that he will have enough to buy the car? (Assume no effect

on inflation).

4. How many years will it take for an initial investment of $10000

to grow to $25000? Assume a rate of interest of 6% compounded

continuously.

5. A stock exchange market is growing continuously at a rate of 5%

per year. How long would it take to double its value?

6. Due to deflation, the purchasing power of your money in the

bank account is decreasing continuously at a rate of 2% per year.

How long would it take to lose 10% of its value?

7. The size P of a certain insect population at time t (in days) obeys

the equation 0.02500 tP e .

a) What is the growth rate?

b) What is the population after 5 days?

c) After how many days will the population reach 1000?

d) The formula P = 500(r)t can be used to calculate the population

after t days. Determine r. (4 decimal places)

8. The number of bacteria N present in a culture at time t (in hours)

follows 0.0131240 tN e .

a) How many bacteria will be in the culture after a day?

b) At what time (in hours) will the population exceed 1500?

9. Strontium-90 is a radioactive material that decays according to

the equation 0.0244t

iA A e , where Ai is the initial amount

present and A is the amount present at time t (in years).

a) What is the half-life of strontium-90?

b) Determine how long it takes for 100 grams of strontium-90 to

decay to 10 grams.

10. The temperature, T, in degrees Celsius, of a cup of coffee t

minutes after it is poured is given by 0.0595 tT e .

a) How hot was the coffee when it was first poured?

b) Find the temperature of the coffee 10 min later.

11. The intensity of light, I, passing through a glass with an

absorption coefficient of 0.2 is given by 0.2

0( ) tI t I e , where I0 is

the initial intensity, and t is the thickness of the glass in centimetres.

What thickness will reduce the intensity to half the initial intensity?

12. The annual rate of a GIC is 6%. Determine its effective rate if it

compounds continuously.

Answers

1a) $344.00 1b) $117.02 2a) $886.92 2b) $654.98

3) $12910.62 4) 15.3 years 5) 13.9 years 6) 5.3 years 7a) 2% 7b) 552 only 7c) 34.7 days 7 d) 1.0202

8a) 1694 8b) 15 hrs 9a) 28.4 yrs 9b) 94.4 yrs

10a) 95oC 10b) 58oC 11) 3.47 cm 12) 6.18%