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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 • 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
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
Properties of exponential functions • b>0 then domain of bx is all real numbers x • Range is (0,+) (if b1) • bx is differentiable • Power laws:
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 = (- , + )
Concrete Examples • Compute the following logarithms
Key facts about logs • logbx only defined for x>0 • Law of logs:
Common logarithms • Usually deal with log10 x= log x or loge x = ln x • Number e:
Equations involving logs, exp • Example: Solve ln (x+1)=5 • Example: Solve ex-3e-x =2 • Example: Rewrite
Logarithms in Science • Richter scale: • M = Magnitude of earthquake in Richters • E = Released energy (joules) • Richter formula:
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?
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%?
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
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
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.
Examples • Compute the derivatives of the following functions:
Logarithmic Differentiation • General Strategy: Differentiate complicated function y = f(x) by simplifying and (implicitly) differentiating both sides of ln |y| = ln |f(x)|
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