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Exponents

Exponential Equations I

Exponential Functions

The Exponential Function Base e

The Logarithm Function

Logarithmic & Exponential Equations

Applications:Growth & Decay

Review&Test

 

jdsmathnotes

 

 


 UNIT 5 : EXPONENTIAL & LOGARITHMIC FUNCTIONS

 LESSON 1: LAWS OF EXPONENTS

 

Examples of Powers:

25 = 2 x 2 x 2 x 2 x 2 = 32; (-3)3 = (-3) x (-3) x (-3) = -27; 2.72 = 2.7 x 2.7 = 7.29

 

A POWER (am ) consists of two parts; the base a and the exponent m.

 

Review of Basic Exponent Laws:

 

Rule

Example

Explanation

am x an = am+n

32 x 35 = 37

Multiplication Rule - If the bases are the same, add the exponents

Division Rule - If the bases are the same, subtract the exponents

(am)n = amn

(32)3=36

Power Rule When taking a power of a power, multiply the exponents

(ab)m = amam

(3 x 2)4 = 34 x 24

Power of a Product Take each factor in the product to that power

Power of a Quotient Take numerator and denominator to that power

 

 

Example 1: Simplify.

 

Solutions:

 

 

Example 2: Simplify.

 

Solutions:

 

 

 

Example 3: Simplify each of the following:

 

Solutions:

 

 

 

Alternate solution for c):

 

 

 

Rational Exponents:

 

Review of Radicals and Roots:

Example 2:

Text Box: Key Idea:
To find the 5th root of 32, work backwards  what number taken to the exponent 5 will yield 32??  25 = 32 and hence  .
To find the 4th  root of 81, work backwards  what number taken to the exponent 4 will yield 81??   34 = 81 and hence   .

 

Text Box: Key Points:
	For fractional exponents, the denominator n gives the index of the root
	The numerator m gives the exponent.
	If n is an odd number, then x can be any real number, positive or negative.
	If n is even, then x must be positive if we are working in the real number system

 

 

 

 

 

 

 

 

 

 

 

Example 4: Simplify

 

 

Solutions:

 

 

 

Example 5: Simplify

Solutions:

 

 

 

Example 6: Simplify

 

Solutions:

 

 

 

 

 

 

 

 

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