Logarithmic Properties Exponential Function y = bx Logarithmic Function x = by y = logbx Exponential Form Logarithmic Form

Exponential Form Logarithmic Form The logarithm is the exponent to which a base must be raised to give a power. Exponential Form Logarithmic Form 53 = 125  72 = 49  5-2 = 1/25  3-4 = 1/81  9½ = 3  36½ = 6  be = P 

Exponential Form Logarithmic Form The logarithm is the exponent to which a base must be raised to give a power. Exponential Form Logarithmic Form 53 = 125  72 = 49  5-2 = 1/25  3-4 = 1/81  9½ = 3  36½ = 6  be = P  3 = log5125 2 = log749 -2 = log5(1/25) -4 = log3 (1/81) ½ = log93 ½= log366 e = logbP

Exponential Form Logarithmic Form The logarithm is the exponent to which a base must be raised to give a power. Exponential Form Logarithmic Form  4 = log381  2 = log864  -2 = log7(1/49)  -4 = log2 (1/16)  ¼ = log813  1/5 = log322  y = logax

Exponential Form Logarithmic Form The logarithm is the exponent to which a base must be raised to give a power. Exponential Form Logarithmic Form  4 = log381  2 = log864  -2 = log7(1/49)  -4 = log2 (1/16)  ¼ = log813  1/5 = log322  y = logax 34 = 81 82 = 64 7-2 = 1/49 2-4 = 1/16 81¼ = 3 321/5 = 2 ay = x

  Method #1 Method #2  

Let 𝑦= 𝑙𝑜𝑔 2 32 𝑙𝑜𝑔 2 32= 𝑙𝑜𝑔 2 2 5 ∴ 𝑙𝑜𝑔 2 32=5 2 𝑦 =32 2 𝑦 = 2 5 𝑦=5   Method #1 Method #2 Write the Argument as a Power of the base of the log Let 𝑦= 𝑙𝑜𝑔 2 32 𝑙𝑜𝑔 2 32= 𝑙𝑜𝑔 2 2 5 Change to Exponential Form The Logarithm is the Exponent ∴ 𝑙𝑜𝑔 2 32=5 2 𝑦 =32 2 𝑦 = 2 5   Equate Exponents 𝑦=5 ∴ 𝑙𝑜𝑔 2 32=5

  Ex #2] Evaluate. a) 𝑙𝑜𝑔 3 1 81 b) 𝑙𝑜𝑔 5 1 c) 𝑙𝑜𝑔 1 000 000

Ex #2] Evaluate. = 𝑙𝑜𝑔 5 5 0 = 𝑙𝑜𝑔 3 3 −4 = 𝑙𝑜𝑔 10 10 6 = 0 = - 4 = 6   Ex #2] Evaluate. a) 𝑙𝑜𝑔 3 1 81 b) 𝑙𝑜𝑔 5 1 c) 𝑙𝑜𝑔 1 000 000 = 𝑙𝑜𝑔 10 1 000 000 = 𝑙𝑜𝑔 3 3 −4 = 𝑙𝑜𝑔 5 5 0 = 𝑙𝑜𝑔 10 10 6 = 0 = - 4 = 6

  Method #1 Method #2  

Let 𝑦= 𝑙𝑜𝑔 3 81 = 3 𝑙𝑜𝑔 3 3 4 3 𝑙𝑜𝑔 3 81 = 3 4 3 𝑦 =81 =81 3 𝑦 = 3 4   Method #1 Method #2 Let 𝑦= 𝑙𝑜𝑔 3 81 Write the argument of the Log as a Power 3 𝑙𝑜𝑔 3 81 = 3 𝑙𝑜𝑔 3 3 4 Change to Exponential Form = 3 4 3 𝑦 =81 =81 3 𝑦 = 3 4 Equate Exponents 𝑦=4   ∴ 3 𝑙𝑜𝑔 3 81 = 3 4 ∴ 3 𝑙𝑜𝑔 3 81 =81

  Ex#4] Evaluate. a) 4 𝑙𝑜𝑔 4 64 b) 7 𝑙𝑜𝑔 7 125 c) 6 𝑙𝑜𝑔 4 16

Ex#4] Evaluate. a) 4 𝑙𝑜𝑔 4 64 b) 7 𝑙𝑜𝑔 7 125 c) 6 𝑙𝑜𝑔 4 16 =64 =125   Ex#4] Evaluate. a) 4 𝑙𝑜𝑔 4 64 b) 7 𝑙𝑜𝑔 7 125 c) 6 𝑙𝑜𝑔 4 16 =64 =125 = 6 𝑙𝑜𝑔 4 4 2 = 6 2 =36

Your Turn… Read: “Example#2” on p.461-462 “In Summary” on p.465 Do: p.466-468 #1cdf,2bde,3,4,5aef,6,9,10,19,20a Corrections: #4d)approx 1.40