Unit 8 [7-3 in text] Logarithmic Functions Today’s Objective: I can write and evaluate logarithmic expressions.

Solving exponential equations using common bases 4 𝑥 = 1 16 4 𝑥 =32 5 𝑥 =125 2𝑥 2 = 2 5 Write each side with the same base. 4 𝑥 = 1 4 2𝑥=5 2 5 𝑥 = 5 3 𝑥= 5 2 4 𝑥 = 4 −2 Since bases are the same exponents must be equal. 2 𝑥 =7 𝑥=−2 𝑥=3

Exponential & Logarithm Equations Inverse Exponential Equation Logarithm Equation 𝑏 log 𝑏 𝑎 = 𝑎 Exponent/power log 𝑏 𝑏 𝑎 = 𝑎 Base Result Common Log: Logs with a base 10 Written log only – no base # needed Calculator button: [LOG] Read: log base b of a

Solving using logs – any base

𝑎= 𝑏 𝑥 ↔ log 𝑏 𝑎 =𝑥 Write in logarithmic form Write in exponential form 14= 5 𝑥 log 5 14 =𝑥 log 14 5 log 3 8 =𝑥 𝑥 3 𝑥 =8 3 23= 𝑒 𝑥 log 𝑒 23 =𝑥 log 5 34 =2𝑥 5 2𝑥 =34 6= 10 3𝑥+1 log 8 (6) +1=𝑥 log 10 6 =3𝑥+1 log 8 6 =𝑥−1 log 6 =3𝑥+1 8 𝑥−1 =6

Solve each equation by using common base method 𝑎= 𝑏 𝑥 ↔ log 𝑏 𝑎 =𝑥 𝑥= log 5 625 log 4 64 =2𝑥+3 log 3 27 =𝑥−7 5 𝑥 =625 3 𝑥−7 =27 4 2𝑥+3 =64 3 3 𝑥−7 = 3 5 𝑥 = 5 4 3 4 2𝑥+3 =4 𝑥−7=3 2𝑥+3=3 𝑥=4 𝑥=0 𝑥=10

Find the inverse of each function 𝑎= 𝑏 𝑥 ↔ log 𝑏 𝑎 =𝑥 𝑦= 5log 7 (𝑥+4) 𝑦= 3 𝑥 4 𝑦= 3 4𝑥 +2 𝑥= 5log 7 (𝑦+4) 𝑥= 3 4𝑦 +2 𝑥 5 = log 7 (𝑦+4) 𝑥= 3 𝑦 4 𝑥−2= 3 4𝑦 log 3 (𝑥−2) =4𝑦 7 𝑥 5 =𝑦+4 4𝑥= 3 𝑦 log 3 (𝑥−2) 4 =𝑦 7 𝑥 5 −4=𝑦 log 3 4𝑥 =𝑦