Lesson 9 – 3 Logarithmic Functions Pre-calculus

Learning Objective To solve log equations Graph log functions To evaluate log expressions

Logarithmic Function 𝑓 𝑥 = 𝑏 𝑥 is one–to–one so it has an inverse Logarithmic Function – The inverse of an exponential function For positive real numbers 𝑥 and 𝑏, 𝑏>0 and 𝑏≠1, 𝑦= 𝑙𝑜𝑔 𝑏 𝑥 iff 𝑥= 𝑏 𝑦 D = 𝑥>0 D =ℛ inverses R = ℛ R =𝑦>0 Do “around the world” start with base 𝑙𝑜𝑔 𝑏 𝑥=y ⇒ 𝑏 𝑦 =𝑥

Logarithmic Equation Solve for x 1. 𝑙𝑜𝑔 7 𝑥=2 7 2 =𝑥 49=𝑥 1. 𝑙𝑜𝑔 7 𝑥=2 7 2 =𝑥 49=𝑥 2. 𝑙𝑜𝑔 2 1 16 =𝑥 2 𝑥 = 1 16 2 𝑥 = 2 −4 𝑥=−4 3. 𝑙𝑜𝑔 𝑥 81=4 ( ) 1 4 ( ) 1 4 𝑥 4 =81 𝑥=3

Solve for x Check – up 1. 𝑙𝑜𝑔 𝑥 125=−3 𝑥 −3 =125 𝑥= 1 5

Graph. Find Domain, Range, x– & y–int, asymptote, increasing, or decreasing. Logarithmic Function 4. 𝑓 𝑥 = 𝑙𝑜𝑔 2 𝑥  𝑦= 𝑙𝑜𝑔 2 𝑥 𝑥 𝑦 2 𝑦 =𝑥 2 −2 = 1 4 −2 −1 1 2 2 −1 = 1 2 Plug values into y 2 0 =1 2 1 =2 2 2 =4 D: 𝑥>0 R: ℛ x–int: (1, 0) y–int: none Asym: x = 0 increasing

Graph. Find Domain, Range, x– & y–int, asymptote, increasing, or decreasing. Logarithmic Function 5. 𝑓 𝑥 = 𝑙𝑜𝑔 1 2 𝑥  𝑦= 𝑙𝑜𝑔 1 2 𝑥 𝑥 𝑦 1 2 𝑦 =𝑥 2 2 =4 −2 −1 1 2 Plug values into y 2 1 =2 2 0 =1 2 −1 = 1 2 2 −2 = 1 4 D: 𝑥>0 R: ℛ x–int: (1, 0) y–int: none Asym: x = 0 decreasing

Logarithmic Function Summary: In the function 𝑓 𝑥 = 𝑏 𝑥 if 𝑏>1, 𝑓(𝑥) increases if 0<𝑏<1, 𝑓(𝑥) decreases

Graph. Find Domain, Range, x– & y–int, asymptote, increasing, or decreasing. Check – up 2. 𝑦= 𝑙𝑜𝑔 4 𝑥 D: 𝑥>0 R: ℛ x–int: (1, 0) y–int: none Asym: x = 0 decreasing

The base of a log function can be any positive number except 1. Basic Log Facts But, there are two popular & powerful common bases. Common Log: 𝒍𝒐𝒈 𝟏𝟎 𝒙  𝐥𝐨𝐠 𝒙 Natural Log: 𝒍𝒐𝒈 𝒆 𝒙  𝐥𝐧 𝒙 These have MANY applications to science & engineering (we’ll see tomorrow) Basic Log Facts: 𝑙𝑜𝑔 𝑏 𝑏 𝑥 =𝑥 𝑙𝑜𝑔 𝑏 𝑏=1 𝑏 𝑙𝑜𝑔 𝑏 𝑥 =𝑥 𝑙𝑜𝑔 𝑏 1=0

Logarithmic Equation Simplify Each Expression 6. 𝑙𝑜𝑔 6 1 9. log 1 100 6. 𝑙𝑜𝑔 6 1 9. log 1 100 =0 (log fact!) = 𝑙𝑜𝑔 10 10 −2 =−2 7. 𝑙𝑜𝑔 3 81 = 𝑙𝑜𝑔 3 3 4 10. 𝑙𝑜𝑔 2 (−8) =4 Undefined Why??? 8. ln 𝑒 3 2 𝑥 =−8 not possible!! = 𝑙𝑜𝑔 𝑒 𝑒 3 11. 𝑒 ln 6 =3 =𝑒 𝑙𝑜𝑔 𝑒 6 =6

Logarithmic Equation Simplify or Solve Each Expression 12. ln 1 𝑒 7 3𝑥+15=12 3𝑥= -3 −7 𝑥=−1 14. 𝑙𝑜𝑔 5 5 3𝑥−2 =19 3𝑥−2=19 3𝑥=21 𝑥=7

Solve Check – up 3. ln 𝑒 7𝑥−3 =25 𝑥=4

Lesson 9–3 Logarithmic Functions WS Assignment Lesson 9–3 Logarithmic Functions WS