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Section 6.3 – Exponential FunctionsLaws of Exponents
If s, t, a, and b are real numbers where a > 0 and b > 0, then:
𝑎𝑠 ∙𝑎𝑡=𝑎𝑠+𝑡 (𝑎𝑠)𝑡=𝑎𝑠𝑡 (𝑎𝑏)𝑡=𝑎𝑡𝑏𝑡
1𝑡=1 𝑎0=1 𝑎−𝑡= 1𝑎𝑡=( 1
𝑎 )𝑡
Definition:
An Exponential Function is in the form,
“a” is a positive real number and does not equal 1
“C” is a real number and does not equal 0
The domain of f(x) is the set of all real numbers
“a” is the base and is the Growth Factor
“C” is the Initial Value because
𝑓 (𝑥+1)𝑓 (𝑥 )
=𝑎→𝐶𝑎𝑥+1
𝐶𝑎𝑥 =𝐶𝑎𝑥𝑎1
𝐶𝑎𝑥 =𝑎
Section 6.3 – Exponential FunctionsExamples
𝑓 (0 )=1 , h𝑡 𝑒𝑟𝑒𝑓𝑜𝑟𝑒𝐶=1
𝑥 𝑓 (𝑥) 𝑓 (𝑥+1)𝑓 (𝑥 )
=𝑎
−1
0
1
2
3
23
132
94
278
12/3
=32
3/21
=32
9/ 43 /2
=32
27/ 89/4
=32
𝑎=32
𝑓 (𝑥 )=𝐶𝑎𝑥
𝑓 (𝑥 )=1( 32 )
𝑥
¿ ( 32 )
𝑥
Section 6.3 – Exponential FunctionsExamples
𝑓 (0 )=1/ 4 , h𝑡 𝑒𝑟𝑒𝑓𝑜𝑟𝑒𝐶=1/4
𝑥 𝑓 (𝑥) 𝑓 (𝑥+1)𝑓 (𝑥 )
=𝑎
−1
0
1
2
3
12
14
18
116
132
1/41/2
=12
1/81/4
=12
1/161/8
=12
1/321/16
=12
𝑎=12
𝑓 (𝑥 )=𝐶𝑎𝑥
𝑓 (𝑥 )=14 ( 3
2 )𝑥
Section 6.3 – Exponential FunctionsProperties of the Exponential Function,
The domain is the set of all real numbers.
The range is the set of all positive real numbers.
The y-intercept is 1; x-intercepts do not exist.
The x-axis (y = 0) is a horizontal asymptote, as x.If a > 1, the f(x) is increasing function.
The graph contains the points (0, 1), (1, a), and (-1, 1/a).
The graph is smooth and continuous.
Section 6.3 – Exponential Functions
𝑓 (𝑥 )=𝑎𝑥 ,𝑎>1The graph of the exponential function is shown below.
𝑦= 𝑓 (𝑥 )=𝑎𝑥
a
Section 6.3 – Exponential FunctionsProperties of the Exponential Function,
The domain is the set of all real numbers.
The range is the set of all positive real numbers.
The y-intercept is 1; x-intercepts do not exist.
The x-axis (y = 0) is a horizontal asymptote, as x.If 0 < a < 1, then f(x) is a decreasing function.
The graph contains the points (0, 1), (1, a), and (-1, 1/a).
The graph is smooth and continuous.
Section 6.3 – Exponential Functions
𝑓 (𝑥 )=𝑎𝑥 ,0<𝑎<1The graph of the exponential function is shown below.
𝑓 (𝑥 )=𝑎𝑥 ,0<𝑎<1
Section 6.3 – Exponential FunctionsEuler’s Constant – e
The value of the following expression approaches e,
(1+ 1𝑛 )
𝑛
as n approaches .
Using calculus notation,
Growth and decay
Compound interest
Differential and Integral calculus with exponential functions
𝑒= lim𝑛→∞ (1+ 1
𝑛 )𝑛
Applications of e
Infinite series
Section 6.3 – Exponential Functions
Solving Exponential Equations
1)
2)
2 𝑥=5
𝑥=52
3)
2 𝑥+4=3
𝑥=−12
Theorem
If , then .
Section 6.3 – Exponential FunctionsSolving Exponential Equations
4)
5)
−4 𝑥=3
𝑥=−34
34 ∙3−4 𝑥− 4=33
Section 6.4 – Logarithmic FunctionsThe exponential and logarithmic functions are inverses of each other.
The logarithmic function is defined by
𝑦=𝑙𝑜𝑔𝑎𝑥𝑖𝑓 𝑎𝑛𝑑𝑜𝑛𝑙𝑦 𝑖𝑓 𝑥=𝑎𝑦
The domain is the set of all positive real numbers .
The range is the set of all real numbers .
The x-intercept is 1 and the y-intercept does not exist.
The y-axis (x = 0) is a vertical asymptote.
If 0 < a < 1, then the logarithmic function is a decreasing function.
The graph contains the points (1, 0), (a, 1), and (1/a, –1).
The graph is smooth and continuous.
If a > 1, then the logarithmic function is an increasing function.
Section 6.4 – Logarithmic Functions
(𝑎 ,1 )
𝑎
𝑦=𝑙𝑜𝑔𝑎𝑥
The graph of the logarithmic function is shown below.
The natural logarithmic function
𝑦=𝑙𝑜𝑔𝑒𝑥=ln 𝑥𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝑥=𝑒𝑦
The common logarithmic function
𝑦=𝑙𝑜𝑔10 𝑥=𝑙𝑜𝑔𝑥 𝑖𝑓 𝑎𝑛𝑑𝑜𝑛𝑙𝑦 𝑖𝑓 𝑥=10𝑦
Section 6.4 – Logarithmic Functions
𝑦= 𝑓 (𝑥 )=𝑎𝑥
a
(𝑎 ,1 )
𝑎
𝑦=𝑙𝑜𝑔𝑎𝑥
Graphs of
Section 6.4 – Logarithmic Functions
𝑦= 𝑓 (𝑥 )=𝑎𝑥
a (𝑎 ,1 )
𝑎
𝑦=𝑙𝑜𝑔𝑎𝑥
Graphs of
Inverse Functions:
Section 6.4 – Logarithmic FunctionsGraphs of
Inverse Functions:
Section 6.4 – Logarithmic Functions
𝑦= 𝑓 (𝑥 )=𝑎𝑥
a (𝑎 ,1 )
𝑎
𝑦=𝑙𝑜𝑔𝑎𝑥
Graph
Section 6.4 – Logarithmic Functions
𝑦= 𝑓 (𝑥 )=𝑎𝑥
a (𝑎 ,1 )
𝑎
𝑦=𝑙𝑜𝑔𝑎𝑥
Graph
Section 6.4 – Logarithmic Functions
𝑦= 𝑓 (𝑥 )=𝑎𝑥
a (𝑎 ,1 )
𝑎
𝑦=𝑙𝑜𝑔𝑎𝑥
Graph
Section 6.4 – Logarithmic Functions
𝑦= 𝑓 (𝑥 )=𝑎𝑥
a (𝑎 ,1 )
𝑎
𝑦=𝑙𝑜𝑔𝑎𝑥
Graph
Section 6.4 – Logarithmic FunctionsChange the exponential statements to logarithmic statements
𝑎5=6.75=𝑙𝑜𝑔𝑎6.7
8𝑥=9.2𝑥=𝑙𝑜𝑔8 6.7
𝑒3=𝑏3=ln𝑏
𝑒𝑥=4𝑥=ln 4
Change the logarithmic statements to exponential statements
5=3𝑥
𝑙𝑜𝑔3 5=𝑥7=𝑥4
𝑙𝑜𝑔𝑥7=4
𝑎=𝑒6
ln 𝑎=6
Solve the following equations𝑙𝑜𝑔3 (2𝑥 )=1
2 𝑥=31
𝑥=32
𝑙𝑜𝑔2(5 𝑥+1)=4
5 𝑥+1=24
5 𝑥+1=165 𝑥=15𝑥=3
𝑙𝑜𝑔2 (32 )=−3 𝑥+9
32=2−3 𝑥+9
5=−3 𝑥+9−4=−3𝑥43=𝑥
25=2−3 𝑥+9
Section 6.4 – Logarithmic FunctionsSolve the following equations
𝑒𝑥=10𝑥=ln 10
𝑒7𝑥=157 𝑥=ln15
𝑥=ln 15
7
8+2𝑒𝑥=12
𝑥=ln 2𝑒𝑥=2
2𝑒𝑥=4
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