2. Aim: What is the natural logarithms? Do Now: HW: p.338 # 8,16,20,26,30,38,42,48,50,52,56,58 Given f(x) = e x write the inverse function

3. x = e y, we can rewrite x = e y as log form log e x = log e e y y = log e x f -1 (x) = log e x is the inverse to f(x) = e x f(x) = log e x is called natural logarithms, the notation is f(x) = ln x, or y = ln x log e x = ln x Natural logarithms is the log with natural base (base e)

4. f(x) = e x y = x f(x) = ln x

5. Properties of natural logarithms 1. ln e x = x 2. e ln x = x 3. ln 1 = 0, e 0 = 1 4. ln e = 1, e 1 = e 5. If ln x = ln y, then x = y

6. Evaluate the following without calculator a. – 1 b. ln e 2 2 c. ln e 0 0 d. 2 ln e 2

7. Simplify: a. –1 + ln e 2x b. = 2x – 1 = x 3 – 8 c. 5 – ln e 2x + e ln x = 5 – 2x + x = 5 – x

8. Evaluate the following to the nearest hundredth: 1. 2. 3. 0.19 0.55 -0.08

9. Find x to the nearest tenth x  17.7 1. ln x = 2.87534 2. ln x = -0.05729 x  0.9 3. e x = 35 x = ln 35  3.6

10. If ln 2 = x and ln 3 = y. Find the #1 & #2in terms of x and y 1. ln 6 2. x + y – y 3. Write as a single logarithms