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Logarithm and Exponential Functions

Logarithm and Exponential Functions. Overview of logs and exponential functions “Logarithm is an exponent” Inverse functions Log functions and exponential functions are inverses of one another Properties of logarithms Logarithms/exponentials in scientific and real-life problems

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Logarithm and Exponential Functions

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  1. Logarithm and Exponential Functions • Overview of logs and exponential functions • “Logarithm is an exponent” • Inverse functions • Log functions and exponential functions are inverses of one another • Properties of logarithms • Logarithms/exponentials in scientific and real-life problems • Derivatives of log and exponential functions

  2. Exponential functions • Let b>0. Then bx well defined if x an integer, rational number. OK if x is irrational by continuity. (Filling in holes) • Different behavior if b>1 or 0<b<1

  3. Properties of exponential functions • b>0 then domain of bx is all real numbers x • Range is (0,+) (if b1) • bx is differentiable • Power laws:

  4. Logarithm is an exponent • Key fact: logbx is unique number so that • Summary: logbx and bx are inverses • Domain logb x = Range bx = (0, +) • Range logb x = Domain bx = (- , + )

  5. Concrete Examples • Compute the following logarithms

  6. Key facts about logs • logbx only defined for x>0 • Law of logs:

  7. Common logarithms • Usually deal with log10 x= log x or loge x = ln x • Number e:

  8. Equations involving logs, exp • Example: Solve ln (x+1)=5 • Example: Solve ex-3e-x =2 • Example: Rewrite

  9. Logarithms in Science • Richter scale: • M = Magnitude of earthquake in Richters • E = Released energy (joules) • Richter formula:

  10. Earthquake questions • Find a formula for E (energy released) • Question: If energy of one earthquake is 10 times greater, then what’s difference in Richters? • Concrete Examples: 1994 Northridge was 6.8. Approximately how much stronger • Kobe 7.2? • 1906 San Francisco 7.8? • 1964 Alaska 8.4?

  11. Real life exponential problem • Receive $10000 at age 20 and invest in mutual fund until retire @ age 70. • How much money if fund earns • 10%? • 15%?

  12. Financial Analysis

  13. Derivative of exponential functions

  14. The exponential function, exp (x)

  15. Examples

  16. More Examples

  17. Inverse Functions • Example: f(x)=3x+1, g(x)=(x-1)/3 • f(g(x))=3[(x-1)/3]+1=x • More generally, say f and g are inverse functions if • f(g(x))=x for all x in domain of g • g(f(x))=x for all x in domain of f • In this case, write g=f -1

  18. Differentiating Inverse Functions • Find the derivative of y = f -1(x) • Step 1: Apply f to both sides to get x = f(y) • Step 2: Differentiate • Step 3: Conclude that

  19. Example • Consider y = f(x) = x13 + 2x + 5. • Compute the derivative of its inverse x = f -1(y) using above formula. • Compute the derivative of its inverse using implicit differentiation.

  20. Inverse tangent, arctan x

  21. Derivative of log functions

  22. Examples • Compute the derivatives of the following functions:

  23. Logarithmic Differentiation • General Strategy: Differentiate complicated function y = f(x) by simplifying and (implicitly) differentiating both sides of ln |y| = ln |f(x)|

  24. Derivatives of irrational powers of x • Let y = xr , x>0, where r is a real number • Differentiate ln y = ln xr • Conclude that for all real numbers have power law

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