Growth: interest, births Decay: isotopes, drug levels, temperature Scales: Richter, pH, decibel levels

Exponential and Logarithm Functions

Functions involved are called exponential and logarithmic.

A function when the base(a) is some positive number.

The exponent is variable(x).

The exponential function with base a is defined by:

Exponential Functions

Example 1x -2 -1 0 1 2

f(x) 1/4 1/2 1 2 4xxf 2)(

-1 1 2

2

4

x

f (x)

Domain:

Range:

Horizontal Asymptote:

x -2 -1 0 1 2

f(x) 9 3 1 1/3 1/9

x

xf

3

1)(

-1 1 2

3

9

x

f (x)

Domain:

Range:

Horizontal Asymptote:

Example 2

Special base, e 2.7182818……..

xexf )(

Use a calculator to evaluate the following values of the natural exponential function (round to 5 decimal places):

Natural Base, e

Exponential functions f (x) = ax are one-to-one functions.

This means they each have an inverse function, a function that reverses what the original function did.

We denote the inverse function with loga, the logarithmic

function with base a, written as:

Logarithmic Functions

and we say “f of x is the logarithm of x base a”.

Exponential vs. logarithmic form

xayx ya log

Switch from logarithmic form to exponential form:

29log3

11.log10

3

12log8

xayx ya log

Switch from exponential form to logarithmic form:

12553

?49log7

2

116 4

1

?4log16

Evaluating logarithms

x

yxxf 2log)(

Domain restrictions (from first week):1. No negatives under an even root2. No division by zero3. Only positives inside a logarithm

1

Domain:

Range:

Vertical Asymptote:

Graph

1. logaax = x (you must raise a to the power of x to get ax)

2. alogax = x (logax is the power to which a must be raised to get x)

Both are also the result of composing a function with its inverse.

Properties of logarithms and exponentials

xxxf loglog)( 10

With calculator: 5 log)5( f

02 log)20( f

Common Logarithm (Base 10)

Without calculator: 001 log)100( f

1. log)1(. f

xxxf e lnlog)(

With calculator: 5 ln)5( f

02 ln)20( f

Without calculator: eef ln)( 33 ln)( eef

Natural Logarithm (Base e)

To evaluate other bases on the calculator, use the following change of base formula:

a

b

log

logloga b

a

b

ln

ln

Isolate exponential function and apply logarithm function to both sides of the equation.

Isolate the logarithm function and apply the base to both sides of the equation.

Remember inverse properties and change of base:

Solving equations

xaxa xxa log and alog

xexe xx ln and ln

Example 1

777:Solve x

1525:Solve 34 xe

Example 2

0)7ln(2:Solve x

Example 3

(a) What is the initial number of bacteria?

(b) What is the relative rate of growth? Express your answer as a percentage.

(c) How many bacteria are in the culture after 5 hours? Please round the answer to the nearest integer.

(d) When will the number of bacteria reach 10,000? Please round the answer to the nearest hundredth.

.49.

400e )( is speciescertain a of population Thet

tn

Example 4

.0049540 )( t.etm

(a) How much remains after 60 days?

(b) When will 10 grams remain? Please round the answer to the nearest day.

(c) Find the half-life of polonium-210.

Example 5The mass m(t) remaining after t days from 40 gram sample of polonium-210 is given by: