3.4 Exponential and Logarithmic Equations Properties of Exp. and Log Functions log a a x = x ln e x = x

Solving Exponential Equations Ex.e x = 72 Take the ln of both sides. ln e x = ln 72 x = ln 72 Ex. 4e 2x = 5

Solving an Exponential Equation Ex. 2(3 2t-5 ) - 4 = 11 First, add 4 to each side 2(3 2t-5 ) = 15 Divide by 2 (3 2t-5 ) = 15/2 ln(3 2t-5 ) = ln 7.5 (2t-5) ln3 = ln 7.5 2tln3 - 5ln3 = ln 7.5 2tln3 = 5ln3 + ln 7.5 = 3.417

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

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

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

Ex.ln (x – 2) + ln (2x – 3) = 2 ln x+ means mult. ln (x – 2)(2x – 3) = ln x 2 e to both sides to get rid of ln’s. 2x 2 – 7x + 6 = x 2 x 2 – 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!