Natural Logarithmic Functions, Logarithmic Function_Differentiation and Integration.pptx

Natural Logarithmic Functions, Logarithmic Function, Differentiation and Integration.

1. The Natural Logarithmic Functions and Logarithmic Functions: Differentiation and Integration

2. Natural Logarithmic Function Definition The natural logarithm function is the inverse of the exponential function , and it is written . This is read as “f of x is the natural log of x”.   2

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4. 4 The exponential function tells you how much something has grown given an amount of time, and the natural log gives you the amount of time it takes to reach a certain amount of growth. Example: Suppose you have invested your money into chocolate, with an interest rate of 100% (because who doesn't want to buy chocolate), growing continuously. If you want to see 20 times your initial investment, how long do you need to wait? Answer: The natural logarithm gives you the amount of time. Since , you would only need to wait about 3 years to see 20 times your initial investment.  

5. 5 Properties of natural logarithmic function Product Rule The natural log of the multiplication of x and y is the sum of the ln of x and ln of y. Example:   Quotient Rule The natural log of the division of x and y is the difference of the ln of x and ln of y. Example:  

6. 6 Properties of natural logarithmic function Reciprocal Rule The natural log of the reciprocal of x is the opposite of the ln of x. Example:   Power Rule The natural log of x raised to the power of y is y times the ln of x. Example:  

7. 7 Key Natural Log Properties Scenario ln Property ln of a Negative Number The ln of a negative number is undefined ln of 0 ln(0) is undefined ln of 1 ln(1)=0 ln of Infinity ln(∞)= ∞ ln of e ln(e)=1 ln of e raised to the x power ln( e x ) = x e raised to the ln power e ln (x) =x

8. 8 Problem 1 Evaluate First, we use the quotient rule to get: Next, we use the power rule to get: If you don't have a calculator, you can leave the equation like this, or you can calculate the natural log values:  

9. 9 Problem 2 Evaluate For this problem, we need to remember that: This means the problem simplifies to : , which is our answer  

10. 10 Problem 3 Solve When you have multiple variables within the ln parentheses, you want to make e the base and everything else the exponent of e . Then you'll get ln and e next to each other and, as we know from the natural log rules, . So, the equation becomes Since , Therefore  

11. 11 Since e is a constant, you can then figure out the value of , either by using the e key on your calculator or using e's estimated value of 2.718. Now we'd add 6 to both sides Finally, we'd divide both sides by 5.  

12. 12 Logarithm Rules ln Rules log( xy )=log(x)+log(y) ln(x)(y)= ln(x)+ln(y) log(x/y)=log(x)−log(y) ln(x/y)=ln(x)−ln(y) log (x a )=  a log ( x ) ln(x a )=  a ln( x ) log(10 x )= x ln( e x )= x 10 log(x) = x e ln (x) = x

13. 13 Derivatives of Logarithmic Functions    

14. 14 Problem 1 If , find  

15. 15 Problem 2 If , find  

16. 16 Problem 3 If , find  

17. 17 Simplify  

18. 18 Integration of Logarithmic Functions  

19. 19 Problem 1 Evaluate  

20. 20 Problem 2 Evaluate  

21. 21 Differentiate and simplify the following: 1. 2. 3.   II. Evaluate and simplify the following: 4 . 5 .  

22. Thank You