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Exponential and Logarithmic Functions

Learning Objectives

- Exponential functions.
- Properties of exponentials.
- Logarithmic functions.
- Properties of logarithms.
- Exponential and logarithmic equations.
- Applications.

Exponential Functions

- Exponential function an exponential function is

of the form f(x) bx where bgt0, b ? 1, and x is

any real number. - Domain R.
- Range (0, ?).
- Euler number e 2.71828 is the base of the

natural exponential function ex. - Growth function bx where b gt 1.
- Decay function bx where 0 lt b lt 1.

Exponential Functions

Exponential Functions

- Example f(x) -2 . 3x1 .

Properties of Exponentials

- For a,b gt0, b ? 1 an bn ? a b
- an am ? n m
- The exponential is a one-to-one function.

Logarithmic Functions

- Logarithmic function the logarithmic function is

the inverse of the exponential function. - Logarithmic function of base b f(x) logbx ,

for b ? 1. - f(x) logbx ? f-1(x) bx where b ? 1, and x is

any real number. - a logbc ? ba c, where b ? 1.
- Domain (0, ?) (the range of exp).
- Range R (the domain of exp).

Logarithmic Functions

- General log the general log is logbx . In

Maple logb(x). - Common log the common log is log10x, or log

x.In Maple log10(x), log10(x). - Natural log the general log is logex ln x.In

Maple ln(x), logexp(1)(x).

Logarithmic Functions

Logarithmic Functions

Logarithmic Functions

- Example f(x) -2 . log3(x1) .

Properties of Logarithms

- For a,b gt0, b ? 1 logax logbx ? a b
- logan logam ? n m
- The logarithm is a one-to-one function. logbbx

x - b logb x x
- logb1 0
- logbx ln(x) / ln(b)

Properties of Logarithms

- logb (ac) logb a logb c
- logb (a/c) logb a - logb c
- logb (ac) c logb a
- logb (a) logc a / logc b

Properties of Exponentials and Logarithms

- y logax ? ay x
- ay x ? y logax
- ax ex ln a

Exponential and Logarithmic Equations

- Solve 85x1 182x-3 e ln (8) (5x1) eln(18)

(2x-3) - ln(8) (5x1) ln(18) (2x-3)x (5ln8 2ln18)

-3ln18 ln8)x - (3ln18 ln8) / (5ln8

2ln18)

Exponential and Logarithmic Equations

- Solve log2 8 log2 9 logx 3 log2 (8 . 9)

logx 3 - ln (72) / ln 2 ln3 / ln x
- ln x ln 3 . ln 2 / ln 72x e (ln 3 . ln 2

/ ln 72) 3 ln 2 / ln 2.36 3 ln 2 / ln

2.2.2.3.3

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