Algebra and Trigonometry, Lecture notes of Mathematics

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MAT 101D: ALGEBRA AND

TRIGONOMETRY NOTES

WEEK 4-

DR. (MRS) AGNES ADOM-KONADU

Department of Mathematics University of Cape Coast Cape Coast

November 17, 2023

Outline

Exponential and Logarithmic Functions

Binomial theorem

Sequences and Series

Definition of Natural Exponential Function

The natural exponential function is the function f denoted by f (x) = ex

where e = 2. 718 .... and x is any real number.

Graphs of Exponential Functions

Graphs of y = ax

In the same coordinate plane, sketch the graphs of f (x) = 2 x and g(x) = 4 x^.

Solution

By plotting these points and connecting them with smooth curves, you obtain the graphs shown in Figure below. Note that both graphs are increasing. Moreover, the graph of g(x) = 4 x^ is increasing more rapidly than the graph of f (x) = 2 x

Graphs of y = a−x

In the same coordinate plane, sketch the graphs of F (x) = 2 −x and G(x) = 4 −x

Solution

By plotting these points and connecting them with smooth curves, you obtain the graphs shown in the figure below. Note that both graphs are decreasing. Moreover, the graph of g(x) = 4 −x^ is decreasing more rapidly than the graph of F (x) = 2 −x^.

The Natural Logarithmic Function

The Natural Logarithmic Function

For x > 0, y = ln x if and only if x = ey The function given by

f (x) = logex = ln x

is called the natural logarithmic function.

Properties of Logarithms

Properties of Logarithms

1. loga 1 = 0 because a^0 = 1.
2. logaa = 1 because a^1 = a.
3. logaax^ = x and alogax^ = x.
4. If logax = logay, then x = y

Evaluating Logarithms

Evaluate each logarithm at the indicated value of x. a. f (x) = log 2 x, x = 32 b. f (x) = log 3 x, x = 1 c. f (x) = log 4 x, x = 2 d. f (x) = log 10 x, x = 1001

Solution

a. f ( 32 ) = log 232 = 5 because 2^5 = 32. b. f ( 1 ) = log 31 = 0 because 3^0 = 1. c. f ( 2 ) = log 42 = 12 because 4^1 /^2 =

d. f

1000

= log 10 1001 = −2 because 10−^2 = 1012 = 1001

Using Properties of Logarithms

a. Solve for x : log 2 x = log 23 b. Solve for x : log 44 = x c. Simplify log 55 x d. Simplify 7log^714

Solution

a. Using the One-to-One Property (Property 4), you can conclude that x = 3. b. Using Property 2, you can conclude that x = 1. c. Using the Inverse Property (Property 3), it follows that log 55 x^ = x. d. Using the Inverse Property (Property 3), it follows that 7 log^714 = 14.

Finding The Domain of Logarithmic Function

Find the domain of each function. a. f (x) = ln(x − 2 ) b. g(x) = ln( 2 − x) c. h(x) = ln x^2

Solution

a. ln(x − 2 ) is defined only when x − 2 > 0, it follows that the domain of f is ( 2 , ∞). b. ln( 2 − x) is defined only when 2 − x > 0, it follows that the domain of g is (−∞, 2 ). c. ln x^2 is defined only when x^2 > 0, it follows that the domain of h is all real numbers except x = 0.

Change-of-Base Formula

Let a, b, and x be positive real numbers such that a ̸= 1 and b ̸= 1. Then loga x can be converted to a different base using any of the following formulas.

Base 10

loga x = log log^1010 xa

Base e

loga x = ln ln^ xa

Base b

loga x = log logbb^ xa

Changing Bases Using Common Logarithms

a. log 4 25 = log^10 log 10 4

≈ 1.^39794

b. log 3 17 = log^10 log 10 3

≈ 1.^23045

Changing Bases Using Natural Logarithms

a. log 4 25 = ln^25 ln 4

≈ 3.^21888

b. log 3 17 =

ln 17 ln 3 ≈^

1. 09861 ≈^2.^58.