Exponents and Logarithms.pptx

Exponents & Logarithms Objectives of Learning Evaluate ax in the case when x is positive, negative, a whole number or fraction. Simplify algebraic expressions using the rules of exponents. Evaluate logarithms in simple cases. Use the rules of logarithms to solve equation...

Exponents & Logarithms Objectives of Learning Evaluate ax in the case when x is positive, negative, a whole number or fraction. Simplify algebraic expressions using the rules of exponents. Evaluate logarithms in simple cases. Use the rules of logarithms to solve equations in which the unknown occurs as a power. 1  Exponent Notation In general, if: M = bn we say that bn is the exponential form of M to base b. n is referred to as the power, index or exponent. 2  Rules of Exponents  Rule 1 : am × an = am+n  Rule 3 :(am)n = amn Example: Example: 52 × 53 = 52+3 = 55 (23)-3 = 23×(-3) = 2-9  RuleExample: 2 : am ÷ an = am-n  Rule 4 :(ab)m = am × bm Example: 35 (4.5 × 3)5 = 4.55 × 35 35 2 33 32 3  Rules of Exponents power m  a am nth root  Rule 5 :    m  Rule 7 : m b n b an  am Example: Example: 3 6 36 52  53  3    6  5 5  125 m 1  Rule 8 : a   Rule 0 6:a =1 am Example: Example: 20 = 1 3 1 3  33 (-12)0 = 1 4  Exponents Example 1: (a) Simplify 1 3 (i) x4 x 4 2 3 x y (ii) x 4 y 5  Exponents Example 1: (b) Solve (i) 2x = 16 (ii) 2 a3 25 6  Exponents Ex. 1: (a) Simplify  1 2 3 3 (i) (x y ) 2 (ii) x 3 x2 7  Exponents Ex. 2: (b) Solve x  1 (i)   125  5 3 (ii) x 2 1 16 x2 8  Logarithms The general form of logarithms is: number log b M n exponent Exponential functions base and Logarithmic functions are reverses of each other. n M b 9  Logarithms Notes:  We can omit “10” for the logarithms to base 10. Other bases need to be indicated.  b must be positive such that m must also be positive. Thus, we can’t determine the logarithms of a negative number.  Logarithms to base e is natural logarithms. 10  Rules of Logarithms Rule 1 : logm (a × b) = logm a + logm b Example: log2 (a × 2) = log2 a + log2 2 a  Rule 2 : logm ( b ) = logm a – logm b 2 Example: log4 ( ) = log4 2 – log4 3 3 Example: log (36) = 6 log 3  Rule 3 : logm (b ) = n logm b 2 n 2 11  Rules of Logarithms  Rule 4 : loga a = 1 Example: log2 2 = 1 log4 4 = 1 (if the base = the number) Example: log2 1 = 0 log5 1 = 0  Rule 5 : loga 1 = 0 (if the number is 1) 12  Rules of Logarithms  Rule 6 : b log b r r Example: log 2 5 2 5  Rule 7 : logm b = logn b / logn m Example: log2 4 = log10 4 / log10 2 13  Logarithms Example 2: Simplify log2 40 + 3 log2 6 – log2 135. [Ans: 6] 14  Logarithms Example 3: Solve (i) 2x = 7 (ii) 25x-1 = 3x [Ans: 2.807; 0.293] 15  Logarithms Ex. 3: Simplify without using calculator: log 5 2  log 32  log 100 (i) log 15  log10 37 37 (ii) log 2  2log 2 2  log 2  log 2 70 35 5 [Ans: 2; 5.322] 16  Logarithms Ex. 4: Solve (i) 25(10)2t = 208 (ii) 38 + 12e-0.5t = 208 (iii) log10 x + log10 (x+9) = 1 [Ans: 0.46; -5.302; 1] 17  Don't say you don't have enough time. You have exactly the same number of hours per day that were given to Helen Keller, Pasteur, Michelangelo, Mother Teresa, Leonardo da Vinci, Thomas Jefferson, and Albert Einstein - H. Jackson Brown 18 FP-036 Commerce Math 062