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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 bgt0. Then bx well defined if x an integer,

rational number. OK if x is irrational by

continuity. (Filling in holes) - Different behavior if bgt1 or 0ltblt1

Properties of exponential functions

- bgt0 then domain of bx is all real numbers x
- Range is (0,?) (if b?1)
- 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 xgt0
- 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 (x1)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 whats 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 _at_ age 70. - How much money if fund earns
- 10?
- 15?

Financial Analysis

Derivative of exponential functions

The exponential function, exp (x)

Examples

More Examples

Inverse Functions

- Example f(x)3x1, g(x)(x-1)/3
- f(g(x))3(x-1)/31x
- 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 gf -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.

Inverse tangent, arctan x

Derivative of log functions

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 , xgt0, where r is a real number
- Differentiate ln y ln xr
- Conclude that for all real numbers have power law

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