Engineering Math 1 (COE0001)
Grachel C. Quiñones
Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Disclaimer: The following information is condensed from Louis
Leithold’s College Algebra and Trigonometry.
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Exponential Function with Base b
Definition
If b > 0, b 6= 1, then the exponential function with base b
is the function defined by
f (x) = b x .
Domain: <
Range: <+
Note: If b = 1, then b x becomes 1x and since 1x = 1 for any
x, we have a constant function.
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Illustration
f (x) = 2x
x
1
f (x) =
2
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Natural Exponential Function
Definition
The natural exponential function is the function f defined
by
f (x) = e x
Domain: <
Range: <+
Note: e ≈ 2.7182818. The letter e was chosen because of the
Swiss mathematician and physicist Leonard Euler (1707-1783).
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Exponential Function
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Exponential Growth and Decay
Exponential growth and exponential decay give
mathematical models involving powers of e.
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Exponential Growth
Definition
A function defined by an equation of the form
f (t) = Be kt , t≥0
where B and k are positive constants, is said to describe
exponential growth.
Note: As t → +∞, Be kt → +∞
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Example:
In a particular bacterial culture, if f (t) bacteria are present at
t minutes, then
f (t) = Be 0.04t
where B is a constant. If there are 1500 bacteria present
initially, how many bacteria will be present after 1 hour?
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Example:
The population of a particular town is increasing proportional
to its size. If this rate is 6% and if the population after t years
is P(t), then
P(t) = ke 0.06t
where k is a constant. If the current population is 10, 000,
what is the expected population after (a) 10 years and (b) 20
years?
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Exponential Decay
Definition
A function defined by an equation of the form
f (t) = Be −kt , t≥0
where B and k are positive constants, is said to describe
exponential decay.
Note: As t → +∞, Be −kt → 0+
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Example:
If V (t) pesos is the value of a certain piece of equipment t
years after its purchase, then
V (t) = Be −0.20t
where B is a constant. If the equipment was purchased for
8,000Php, what will be its value in 2 years?
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Example:
The value of a particular machine t years after its purchase is
v (t) pesos, where
v (t) = ke −0.30t
and k is a constant. If the machine was purchased 8 years ago
for 10,000Php, what is its value now?
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Bounded Growth
Definition
Another mathematical model involving powers of e is given by
the function defined by
f (t) = A(1 − e −kt ) = A − Ae −kt .
This function describes bounded growth.
Note: If t = 0, we’ll have that f (0) = 0. As t → +∞,
Ae −kt → 0. This means that as t → +∞, A − Ae −kt → A.
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Example:
A typical worker at a certain factory can produce f (t) units
per day after t days on the job, where
f (t) = 50(1 − e −0.34t ).
a. How many units per day can the worker produce after 7
days on the job?
b. How many units per day can the worker eventually be
expected to produce?
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Logarithmic Functions with Base b
Definition
The logarithmic function with base b is the inverse of the
exponential function with base b.
y = logb x ⇐⇒ x = b y or
b y = x ⇐⇒ logb x = y
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Illustration
32 = 9 ⇐⇒ log3 9 = 2
23 = 8 ⇐⇒
1
5−2 = ⇐⇒
25
log6 1 = 0 ⇐⇒ 60 = 1
log10 10000 = 4 ⇐⇒
1
log8 2 = ⇐⇒
3
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Examples:
log7 49
√
log5 5
1
log6
6
log3 81
log10 0.001
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Examples:
log6 x = 2
2
log27 x =
3
1
logb 4 =
3
logb 81 = −2
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Logarithmic Functions
Notes: For b > 0, b 6= 1, x > 0 and y a real number,
b logb x = x
logb b y = y
logb 1 = 0
logb b = 1
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Natural Logarithmic Function
Definition
The natural logarithmic function is the inverse of the
natual exponential function.
y = loge x = ln x ⇐⇒ x = e y
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Natural Logarithmic Function
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Natural Logarithmic Function
Notes:
ln e = 1
e ln x = x
ln e x = x
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Properties of Logarithmic Functions
If b > 0, b 6= 1, n is any real number, and u and v are positive
numbers, then
logb uv = logb u + logb v
u
logb = logb u − logb v
v
logb u n = n logb u
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Illustration:
log2 4 · 8
128
log2
16
log2 43
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Examples
logb x 2 y 3 z 4
x
logb 2
yz
r
2
5 xy
logb
z3
log10 28
log10 2100
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Exponential and Logarithmic Equations
Note: log x is used to represent log10 x.
log x = y ⇐⇒ 10y = x
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Exponential Functions
Logarithmic Functions
Exponential and Logarithmic Equations
Examples:
Solve the following:
53x−1 = 0.08
7x = 3x+1
log4 19
log10 (x + 3) = 2
log2 (x + 4) − log2 (x − 3) = 3
log3 x + log3 (2x − 3) = 3
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