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Standard level

4.07 The natural logarithm

Worksheet
Euler's number
1

Consider the function f \left( x \right) = e^{x}.

a

Complete the following table of values, correct to two decimal places:

x-3-2-10123
f(x)
b

Sketch the graph of f \left( x \right) = e^{x}.

2

Sketch the curves y = e^{x}, y = e^{x} + 3, and y = e^{x} - 4 on the same coordinate plane.

3

Use a calculator or other technology to approximate the each of the following values correct to four decimal places:

a

e^{4}

b

e^{ - 1 }

c

e^{\frac{1}{5}}

d

5 \sqrt{e}

e

\dfrac{4}{e}

f

\dfrac{8}{9 e^{4}}

The natural logarithm
4

Find the value of each of the following correct to four decimal places:

a

\ln 94

b

\ln 0.042

c

\ln 78^{4}

d

\ln \left( 18 \times 35\right)

5

Consider x=\ln 31. Find the value of x, correct to two decimal places.

6

Find the value of 5 \log_{e} e.

7

Is the value of \log_{e} 2 greater than or less than 1?

8

Evaluate each of the following expressions:

a

\ln e^{3.5}

b

\ln e^{4}

c

\sqrt{6} \ln \left(e^{\sqrt{6}}\right)

d

\ln \left(\dfrac{1}{e^{2}}\right)

9

Write the following equations in logarithmic form.

a
e^{0} = 1
b
e^{x} = 3
c
e^{\frac{1}{2}} = \sqrt{e}
d
e^{p} = q
Applications
10

The population P of a city increases according to the formula P = 3000 e^{ k t} where t is measured in years and k is a constant.

a

Find the initial population.

b

Given the population increases to 8000 in 2 years find the exact value of k.

c

How many complete years will it take for the population to at least double?

11

The voltage V in volts across an electrical component decays according to the equation V = A e^{ - \frac{1}{2} t } where A is the initial voltage and t is the time in years. Find the value of t such that V is half of the initial voltage, correct to two decimal places.

12

The spread of a virus through a city is modelled by the function: N = \dfrac{15\,000}{1 + 100 e^{ - 0.5 t }}, where N is the number of people infected by the virus after t days.

a

How many people will have been infected after 3 days? Round your answer to the nearest whole number.

b

How many whole days will it take for at least 4000 people to be infected with the virus?

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