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Exponential functions and logarithms

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A. Exponential functions and exponential growth

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Example 1: the function y=2x

Table x y

-4 2-4=1/16=0.0625

-3 2-3=1/8=0.125

-2 2-2=1/4=0.25

-1 2-1=1/2=0.5

0 20=1

0.25 20.25=1.1892…

0.5 20.5=1.4142…

0.75 20.75=1.6817…

1 21=2

2 22=4

3 23=8

4 24=16

Graph

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Exponential function versus power function

y=2x describes an exponential function

A power function is a function having an equation of the form y=xr (where r is a real number), i.e. x serves as the base.

An exponential function is a function having an equation of the form y=bx (where b is a positive number distinct from 1), i.e. x is the exponent.

x is the exponent x is the base

y=x2 describes a (quadratic function), power function

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Example 2: a growing capital

An amount of 1000 EUR is invested in a savings account yielding 3% of compound interest each year. Express the amount A in the savings account in terms of the time t (in years, starting from the time of the investment).

t=1: A=1000+0.031000=1000+30=1030

t=2: A=1030+0.031030=1030+30.9=1060.9

t=3:

t=4:

t=5:

A=1060.9+0.031060.9=1060.9+31.82…=1092.72…

A=1092.72…+0.031092.72…=1092.72…+32.78…=1125.50…A=1125.50…+0.03 1125.50…=1125.50…+33.76…=1159.27…

general formula???

in the beginning: 1000 EUReach year: + 3% (of the preceding value)

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Example 2: a growing capital

An amount of 1000 EUR is invested in a savings account yielding 3% of compound interest each year. Express the amount A in the savings account in terms of the time t (in years, starting from the time of the investment).t=1: A=1000+0.031000=1000+30=1030

t=2: A=1030+0.031030=1030+30.9=1060.9

t=3: A=1060.9+0.031060.9=1060.9+31.82…=1092.72…

A=1000+0.031000=1000(1+0.03)=10001.03=1030

A=1030+0.031030=1030(1+0.03)=10301.03 =10001.031.03=10001.032(=1060.9)

A=1060.9+0.031060.9=1060.9(1+0.03)=1060.91.03

=10001.031.031.03 =10001.033(=1092.72…)

each year ×1.03 A=10001.03t

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Example 2: a growing capital

An amount of 1000 EUR is invested in a savings account yielding 3% of compound interest each year. Express the amount A in the savings account in terms of the time t (in years, starting from the time of the investment).

A=10001.03t=

t

100

311000

‘each year: +3%’ corresponds to‘each year ×1.03’ (1.03=1+3/100)

multiple of an exponential function!

we will use this formula also if t is not an integer

graph has J-form

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Example 2: a growing capital

An amount of 1000 EUR is invested in a savings account yielding 3% of compound interest each year. Express the amount A in the savings account in terms of the time t (in years, starting from the time of the investment).

A=10001.03 t=

t

100

311000

growth factor

initial value=1000

growth factor = 1.03

yearly growth percentage=3%

graph has J-form

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Exponential growth

• A variable y grows exponentially iff y=y0bt (y0: initial value; b growth factor (b>0, b≠1))

• If y increases by p% every time unit(p: growth percentage), then

♦ y grows exponentially

♦ growth factor is

♦ the equation is

♦ the graph has J-form

1001

pb

tp

yy

10010

cf. examples 1 and 2

cf. example 2

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Exercise

growth percentage(+ …% each time unit)

growth factor(×… each time unit)

+5% ×1.05

+50% ×1.5

+0.5% ×1.005

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Example 3: decreasing population of a town

A town had 100 000 inhabitants on 1 Jan. 1950, but since then its population decreased by 3% each year. Express the population N in terms of the time t (in years, starting from 1 Jan. 1950).

t=1:

t=2:

t=3:

N=1000-0.031000=1000(1-0.03)=10000.97=970

N=970-0.03970=970(1-0.03) =10000.970.97=10000.972

N=940.9-0.03940.9=940.9(1-0.03)

=10000.973

A=10000.97t

graph has reflected

J-form

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Exponential increase/decrease

• If y decreases by p% every time unit(negative growth percentage), then

♦ y grows exponentially

♦ growth factor is <1:

♦ the equation is

♦ the graph has reflected J-form

• An exponential function y=bx is• increasing if b>1• decreasing if b<1

1001

pb

tp

yy

10010

cf. example 3

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Exercise

growth percentage(+ …% each time unit)

growth factor(×… each time unit)

+5% ×1.05

+50% ×1.5

+0.5% ×1.005

–5% ×0.95

–50% ×0.5

–0.5% ×0.995

+100% ×2

+1000% ×11

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14 A. Exponential functions and exponential growth

HandbookChapter 4: Exponential and logarithmic

functions4.1 Exponential functions• introduction and definition• examples 1, 2, 3, 6 and 7• problems 16, 18, 19, 20, 30, 31, 32, 33, 34, 35,

36

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B. Logarithms

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Example

Find x such that …

100010 x

10010

1000x100010 x

990101000 x

100010 x 3x 1000log

3 is the (common) logarithm (or logarithm base 10) of 1000

in words: which exponent do you need to obtain 1000

when the base of the power is 10?

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Logarithms

(common) logarithm (logarithm base 10) of x:log x = y iff 10y = x

100log

in words: log x is the exponent needed to make a power with base 10 equal to x

Calculate the following logarithms (without calculator)

0000001log

001.0log

10log

1log

10010? 10010!

2 2100log 60000001log

3001.0log 110log 01log

100log 0log

undefined

undefined

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Logarithms using the calculator

Calculate the following logarithms and verify the result

2log

20log3000log

3log

5log6log

4log

8log9log

...029301.0

...029301.1

...121477.0...059602.0...970698.0...151778.0...089903.0...242954.0

...121477.3

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Some rules for calculations with logarithms

2log

20log3000log

3log

5log6log

4log

8log9log

...029301.0

...029301.1

...121477.0...059602.0...970698.0...151778.0...089903.0...242954.0

...121477.3

20log10log2log

3000log1000log3log

...477.03!

...477.03 101010

||||||

310003000

10log5log2log

6log3log2log

Logarithm of a product: baba logloglog

3000log

3log1000log

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Some rules for calculations with logarithms

2log

20log3000log

3log

5log6log

4log

8log9log

...029301.0

...029301.1

...121477.0...059602.0...970698.0...151778.0...089903.0...242954.0

...121477.3

3log29log3log 2

3log3log33log!

2log38log

2log24log

Logarithm of a power: ara r loglog

22log

32log

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C. Exponential equations

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Example 1: a growing capital

An amount of 1000 EUR is invested in a savings account yielding 3% of compound interest each year. Express the amount A in the savings account in terms of the time t (in years, starting from the time of the investment).

When will the amount in the savings account be equal to 1500 EUR?

A=10001.03t

t? such that A=1500

150003.11000 t

5.103.1 t

5.103.1 tlog( ) log( )

5.1log03.1log t

...7.1303.1log

5.1logt

ara r loglog (apply )

exponential equation: unknown is in the exponent

Answer: After about 13.7… years, the amount is equal to 1500 EUR.

(divide by 1000)

(take logarithm of both sides)

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Example 2: two growing capitals

tt 035.190003.11000

log( ) log( )

1000

900

035.1

03.1

t

t

1000

900

035.1

03.1

t

1000

900

035.1

03.1

t

1000

900log

035.1

03.1log t

035.103.1

log

1000900

logt

A=10001.03t

t? such that A=J

J=9001.035t

Ann invests an amount of 1000 EUR in a savings account yielding 3% of compound interest each year. John invests 900 EUR in a savings account yielding 3.5% of compound interest each year.

When will they have the same amount in their savings account?

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Example 2: two growing capitals

035.103.1

log

1000

900log

t ...7.21t

Answer: It takes nearly 22 years before the two amounts are equal.

A=10001.03t

J=9001.035t

Ann invests an amount of 1000 EUR in a savings account yielding 3% of compound interest each year. John invests 900 EUR in a savings account yielding 3.5% of compound interest each year.

When will they have the same amount in their savings account?