Logarithmic Equations Mrs.Volynskaya Pre-Calculus Chapter 3.4 Exponential and Logarithmic Equations Properties of Exp. and Log Functions loga ax = x ln ex = x

Solving Exponential Equations Take the ln of both sides. Ex. ex = 72 ln ex = ln 72 x = ln 72 Ex. 4e2x = 5

Solving an Exponential Equation First, add 4 to each side 2(32t-5) = 15 Divide by 2 (32t-5) = 15/2 ln(32t-5) = ln 7.5 (2t-5) ln3 = ln 7.5 2tln3 - 5ln3 = ln 7.5 2tln3 = 5ln3 + ln 7.5 = 3.417

Ex. e2x – 3ex + 2 = 0 (ex)2 – 3ex + 2 = 0 This factors. ( ) ( ) = 0 ex – 2 ex - 1 Set both = 0 and finish solving. ex = 2 ex = 1 ln ex = ln 2 ln ex = ln 1 x = ln 2 x = 0

Ex. 2x = 10 ln 2x = ln 10 x ln 2 = ln 10 Ex. 4x+3 = 7x ln 4x+3 = ln 7x (x + 3) ln 4 = x ln 7 x ln 4 + 3 ln 4 = x ln 7 Collect like terms 3 ln 4 = x ln 7 - x ln 4 Factor out an x 3 ln 4 = x( ln 7 – ln 4)

Solving a Logarithmic Equation Ex. ln x = 2 Take both sides to the e eln x = e2 x = e2 Ex. 5 + 2 ln x = 4 2 ln x = -1 ln x = Ex. 2 ln 3x = 4 ln 3x = 2 3x = e2

Ex. ln (x – 2) + ln (2x – 3) = 2 ln x + means mult. ln (x – 2)(2x – 3) = ln x2 e to both sides to get rid of ln’s. 2x2 – 7x + 6 = x2 x2 – 7x + 6 = 0 ( ) ( ) = 0 x – 6 x – 1 6 and 1 are possible answers Remember, can not take the log of a neg. number or zero. Put answers back into the original to check them. Notice that only 6 works!