exponential function
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Transcript exponential function
1
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
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Graph
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
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Exponential function versus power function
x is the exponent
y=2x describes an
exponential function
x is the base
y=x2 describes a
(quadratic function),
power 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.
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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).
in the beginning: 1000 EUR
each year: + 3% (of the preceding value)
general formula???
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…
t=4: A=1092.72…+0.031092.72…=1092.72…+32.78…=1125.50…
t=5: A=1125.50…+0.03 1125.50…=1125.50…+33.76…=1159.27…
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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
A=1000+0.031000=1000(1+0.03)=10001.03=1030
t=2: A=1030+0.031030=1030+30.9=1060.9
A=1030+0.031030=1030(1+0.03)=10301.03
=10001.031.03=10001.032(=1060.9)
t=3: A=1060.9+0.031060.9=1060.9+31.82…=1092.72…
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
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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).
‘each year: +3%’ corresponds to
‘each year ×1.03’
(1.03=1+3/100)
we will use this formula
also if t is not an integer
A=10001.03t=
3
1000 1
100
multiple of an
exponential function!
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t
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).
yearly growth percentage=3%
initial value=1000
A=10001.03
3
1000 1
100
t
t=
growth
factor
= 1.03
growth
factor
graph has J-form
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Exponential growth
cf. examples 1 and 2
• 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
cf. example 2
♦ y grows exponentially
♦ growth factor is
♦ the equation is
p
b 1
100
p
y y0 1
100
♦ the graph has J-form
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t
10
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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
A small village had 1000 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: N=1000-0.031000=1000(1-0.03)=10000.97=970
t=2: N=970-0.03970=970(1-0.03)
=10000.970.97=10000.972
t=3: N=940.9-0.03940.9=940.9(1-0.03)
=10000.973
N=10000.97t
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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
♦
♦
p
growth factor is <1: b 1
100
t
p
the equation is y y0 1
100
♦ the graph has reflected J-form
• An exponential function y=bx is
• increasing if b>1
• decreasing if b<1
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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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A. Exponential functions and
exponential growth
Handbook
Chapter 4: Exponential and logarithmic
functions
4.1 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 …
10 x 1000
x 1000 10 990
10 x 1000
1000
x
100
10
10 x 1000
x
in words: which exponent
do you need to obtain 1000
when the base of the
power is 10?
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log1000 3
3 is the (common) logarithm (or
logarithm base 10) of 1000
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Logarithms
in words: log x is the exponent needed to
make a power with base 10 equal to x
(common) logarithm (logarithm base 10) of x:
log x = y iff 10y = x
Calculate the following logarithms (without calculator)
log100
log1 000 000
log 0.001
log10
log1
log 100
log 0
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10 100
?
!
102 100
log100 2
log1 000 000 6
log 0.001 3
log10 1
log1 0
undefined
undefined
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Logarithms using the calculator
Calculate the following logarithms
and verify the result
log 2
log 3
log 4
log 5
log 6
log8
log 9
log 20
log 3000
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0.301 029 ...
0.477 121 ...
0.602 059 ...
0.698 970 ...
0.778 151 ...
0.903 089 ...
0.954 242 ...
1.301 029 ...
3.477 121 ...
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Some rules for calculations with logarithms
log 2
log 3
log 4
log 5
log 6
log8
log 9
log 20
log 3000
0.301 029 ...
0.477 121 ...
0.602 059 ...
0.698 970 ...
0.778 151 ...
0.903 089 ...
0.954 242 ...
1.301 029 ...
3.477 121 ...
log 2 log 3 log 6
log 2 log5 log10
log 2 log10 log 20
log 3 log1000 log 3000
log 3000
3000
||
3 0.477...
10
1000
||
!
103
100.477...
log1000 log 3
Logarithm of a product: loga b log a log b
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3
||
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Some rules for calculations with logarithms
log 2
log 3
log 4
log 5
log 6
log8
log 9
log 20
log 3000
0.301 029 ...
0.477 121 ...
0.602 059 ...
0.698 970 ...
0.778 151 ...
0.903 089 ...
0.954 242 ...
1.301 029 ...
3.477 121 ...
log22 log 4 2 log 2
log23 log8 3 log 2
log32 log9 2 log3
log3 3 log 3 log 3
Logarithm of a power: log a r r log a
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!
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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
exponential equation:
unknown is in the exponent
1000 1.03t 1500 (divide by 1000)
1.03t 1.5
(take logarithm of both sides)
t
log(1.03 )log(1.5)
t log1.03 log1.5
log1.5
t
13.7...
log1.03
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(apply log a r r log a )
Answer: After about 13.7…
years, the amount is equal to
1500 EUR.
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Example 2: two growing capitals
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?
A=10001.03t
J=9001.035t
t? such that A=J
1000 1.03t 900 1.035t
1.03t
900
t
1.035 1000
t
900
1.03
1.035 1000
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t
900
1.03
log
log
1000
1.035
1.03
900
t log
log
1.035
1000
900
log
1000
t
1.03
log
1.035
(
)
(
)
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Example 2: two growing capitals
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?
A=10001.03t
J=9001.035t
900
log
1000
t
1.03
log
1.035
t 21.7...
Answer: It takes nearly 22 years before the two amounts
are equal.
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