(A) Lesson Contet BIG PICTURE of this UNIT: CONTEXT of this LESSON: What are & how & why do we use eponential and logarithic functions? proficiency with algeraic anipulations/calculations pertinent to eponential & logarithic functions proficiency with graphic representations of eponential & logarithic functions Where we ve een Where we are Where we are heading Previous ath courses working with Eponent Laws and graphs of Eponential Relations Reviewing the eponent laws and working with rational eponents Working with Eponential functions in odeling proles and as functions (transforations & inverses) (B) Lesson Ojectives a. Review the asic Eponent Laws (C) Eponent Laws Definition of the ters in an eponential equation: = p is the ase (of the eponent) is the eponent p is the power (the result of repeatedly ultiplying y itself, nuer of ties, or a ase raised to an eponent) Eaple: In = 8, the ase is, the eponent is and the power is 8. This can e read as the following: Two cued is 8. Two to the eponent is 8. Two to the is 8. Eight is the third power of. BUT it CANNOT e read as: Two to the power is 8. (The power is NOT - the power is 8 and the EXPONENT is!) 1 of 5
EXPONENT LAWS: 1. Coparison of ases: If two powers have the sae ases, then their eponents ust e equal. a = if and only if a = ( 0, a > 0, > 0 ). Coparison of eponents: If two powers have the sae eponents, then their ases ust e equal. eponents (with like ases): = y if and only if = y ( 1, 0,1). Multiplication of like ases: When ultiplying ( or ore) like ases, keep the ase and ADD the eponents. y = + y 4. Division of like ases: When dividing like ases, keep the ase and SUBTRACT the eponents. y = y (as long as 0 ) 5. Power of a product: If a single ter is eing raised to an eponent, then the eponent applies to each factor of the single ter. ( a) = a Coon istake: a + ( ) a + (this is NOT TRUE ecause the ase of ( a + ) is not a single ter, ut rather two ters) 6. Power of a quotient: If a fraction is eing raised to an eponent, then the eponent applies to oth the nuerator and the denoinator of the fraction. a, 0 a = a = a, 0 (why can t equal zero?) 7. Power of a power: When a power (such as ) is eing raised to another (outer) eponent, the result is called a power of a power. In this case, keep the ase and ultiply eponents. ( ) y = y of 5
Because the order of ultiplication (coutativity) does not atter, these are equivalent: ( ) y = y and ( ) y = y. 8. Eponent of zero: Any ase raised to an eponent of zero (or the zeroeth power of any ase) is ALWAYS equal to one. 0 = 1 One eception is 0 0 ; this is a non-unique or indeterinate value that arises often in calculus. 9. Negative eponent: When a ase is raised to a negative eponent, reciprocate the ase and raise the result to the positive eponent. = 1, 0 (why can t equal zero?) 10. Fractional eponent: When the eponent of a ase is a fraction, the nuerator of the fractional eponent acts as a regular eponent while the denoinator of the fractional eponent indicates a root of the ase. n = n The syol n is called a radical or n th -root syol. The nuer n in the V is called the inde (or type of root). If no nuer is specified, the type of root is autoatically a SQUARE root. Otherwise, refer to the root as the n th root, as 8 in 56 is the eighth root of 56. You can either work out the ase raised to the eponent first and then take the ( ) root: n = n OR you can work out the n th root of the ase first and then apply the eponent: n = n Advanced lingo: The ase of the eponential epression. eponent on the ase is and the ( th ) power of the ase is the result What are the ase, eponent and power of ( ) n? n is n, the n. of 5
Eercises: 1. Identify the parts of an eponential equation. State the ase, the eponent and the power for each. ( ) = 64 c) e = p e) n a) 4 ) 5 = 1 d) j 0 = 1 f) n. Use the eponent laws to write each epression with a single, siplified ase. a) 4 5 9 c) 1 4 ) 4 5 d) a10 e) a a 5 a 14 f) g 7 = z k = y ( ) k a g) k a ( ) 0 h). Use the eponent laws to write each epression without any zero, negative or fractional eponents. a) w c) 4 5 e) r ) ( a ) a 7 d) 4 5 f) ( ) 1 r 5 ( r ) 4 k 7a 1 + 4 4 5 5 6 6 0 4. Rewrite the following epressions without a fractional eponent (where applicale) and siplify the (resulting) radicals. 1 ( ) a) a c d 10 e 1 ) a 7 6 c 5 d 4 e f 5. Siplify the following epressions so that the final answers contain as few ases as possile ut does not contain zero, negative or fractional eponents. a) 5 y 7 z 10 ( y z ) 4 c) ( ) ) a c 1 c 5 a 6 4 d) 5 6 7 n 1 p 4 q 101 n pq 18 f 4 g h 0 f g 1 4 of 5
6. Siplify the following. a) 1 + 1 ) c) y 4 y 7 9 4 y 4 y 1 ( ) ( a ) 1 h) ( a ) 1 ( a ) 1 d) y e) f) d 10 49 6 0.5 g) 18 4 y z 9 i) ( a + ) 1 7. Evaluate (siplify as a nuer) the following. a) f) 8 1 k) 8 7 1 ) ( ) g) 8 1 l) 8 7 c) h) 8 4 ) ( ) i) 16 4 n) 100 1 61 d) ( ) 1 j) 16 0.5 o) 81 e) + 5 4 0.5 5 of 5