Aim: How do we differentiate the natural logarithmic function?

1. Aim: How do we differentiate the natural logarithmic function? Power Rule Do Now:

2. Exponential example y =2x Inverse of Exponential example x =2y Logarithmic example y = log2x Inverse Exponential Function Exponential Equation y =bx Logarithm = Exponent y = bx “x is the logarithm of y” y = logbx “y is the logarithm of x Inverse of Exponential Equation x =by Logarithmic Equation y = logbx logbx = y if and only if by = x The expression logbx is read as the “log base b of x”. The function f(x) = logbx is the logarithmic function with base b.

3. because e0 = 1 1. ln 1 = 0 2. ln e = 1 because e1 = e 3. ln ex = x because ex = ex inverse property 5. If ln x = ln y, then x = y The logarithmic function with base e is called the natural log function. Natural Logarithmic Function f(x) = logex = ln x, x > 0

4. Properties of Natural Log • The domain is (0, ) and the range is (-, ). • The function is continuous, increasing, and one-to-one. • The graph is concave down. • If a and b are positive numbers and n is rational, then the following are true • ln(1) = 0 • ln(ab) = ln a + ln b • ln(an) = n ln a • ln (a/b) = ln a – ln b

5. Using Properties of Natural Logarithms Rewrite each expression: ln ex = x = -1 ln ex = x = 2 2. ln e2 3. ln e0 ln ex = x because ex = ex = 0 4. 2ln e = 2 ln e = 1 because e1 = e

6. ` ln 6 ln 7/27 Model Problems Use natural logarithms to evaluate log4 30 Given ln 2  0.693, ln 3  1.099, and ln 7  1.946, use the properties of logs to approximate a) ln 6 b) ln 7/27 = ln (2 • 3) = ln 2 + ln 3  0.693 + 1.099  1.792 = ln 7 – ln 27 = ln 7 – 3 ln 3  1.946 – 3(1.099)  -1.351

7. Model Problems Use properties of logarithms to rewrite = ln(3x – 5)1/2 – ln 7 = 1/2 ln(3x – 5) – ln 7

8. Power Rule – the exception

9. Power Rule Power Rule – the exception don’t work!

10. Power Rule The domain of the natural logarithmic function is the set of all positive real numbers. 2nd Fundamental Theorem of Calculus Power Rule – the exception no antiderivative for f(x) = 1/x Definition of the Natural Logarithmic Function accumulation function

11. Definition of the Natural Log Function ln x is negative when x < 1 ln x is positive when x > 1 x 1 x 1 ln(1) = 0 The natural log function measures the area under the curve f(x) = 1/x between 1 and x.

12. e (e, 1) e What is the value of x? x e 1

13. 2nd Fundamental Theorem of Calculus The Derivative of the Natural Log Function Chain Rule

14. Model Problems u’ = 2

15. Model Problems u’ = 2x

16. Rewrite Before Differentiating rewrite u’ = 1

17. Model Problem rewrite u = x2 + 1 u = 2x3 – 1 u’ = 2x u’ = 6x2

18. Logarithmic Differentiation Applying the laws of logs to simplify functions that include quotients, products and/or powers can simplify differentiation. y is always positive therefore ln y is defined take ln of both sides Log properties Differentiate

19. Using Log Derivative Solve for y’, Substitute for y & Simplify

20. Derivative Involving Absolute Value Find the derivative of f(x) = ln|cosx| u = cosx u’ = -sinx

21. Model Problem u = x2 + 2x + 3 u’ = 2x + 2 1st Derivative Test Evaluate critical point f(-1) = ln[-12 + 2(-1) + 3] = ln 2 Relative Extrema – (-1, ln2) Minimum 2nd Derivative Test

22. Aim: How do we differentiate the natural logarithmic function? Do Now: