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Lecture 2:

Topic: Introduction

Reading for this lecture

H&P, Ch 4, Sect 4.2 4.4

Homework for Tutorial in Week 2

Ex 4.2, pp. 180, problems 1, 3, 17, 29.

Ex 4.3, pp. 185-186, problems 11, 21, 25.

Ex 4.4, pp. 190-191, problems 1, 7.

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Logarithmic Functions

Definition

The logarithm of a number, x, to a given base, b,

is the exponent, y, to which the base must be

raised to obtain x.

That is, if, and only if,

where b > 0, b 1, x is any positive real number,

y is any real number.

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From Exponential Form to

Logarithmic Form

102 = 100 implies =

52 = 25 implies =

161/2 = 4 implies =

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From Logarithmic Form to

Exponential Form

log81 = 0 implies =

log232 = 5 implies =

log3(1/9) = -2 implies =

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y = log10x that is, f(x) = log10x

x y

0.5 -0.301

1.0 0.0

1.5 0.176

2.0 0.301

10.0 1.0

100 2.0

Graphing Log Functions, b > 1

x

y

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y = log0.1x that is, f(x) = log0.1x

x y

0.5 0.301

1.0 0.0

1.5 -0.176

2.0 -0.301

10.0 -1.0

100 -2.0

Graphing Log Functions, 0 < b < 1

x

y

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Properties of Logarithms

1. logb(xy) = logb(x) + logb(y)2. logb(x/y)= logb(x) - logb(y)

3. logb(xr) = r.logb(x)

4. logb(1/x)= - logb(x)

5. logb(1) = 0

6. logb(b) = 1

7. logb(br) = r

8. logb(x) = loga(x)/ loga(b)

9. b = xlogbx8

Applying the Properties

Find log0.12 using logs to base 10.

log0.12 = by Property 8

=

=

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Applying the Properties

Find in terms of log10 x and

log10 (x+1)

log10x

(x+1)3= Property 2

= Property 3

log10x

(x+1)3

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The Number e

Euler (1707-1783) discovered the irrational number

e = 2.71828

e equals the value of (1 + x)1/x

when x is very, very close to 0.

e.g. (1 + 0.0001)10,000 =

e.g. (1 + 0.000001)1,000,000 =

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Common Logarithms and

Natural Logarithms

Common logarithms have base 10.

Common logs are written log10x or logx.

Natural logarithms are to base e.

Natural logs are written logex or ln x.

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Single-Valued Functionsy = bx y = logbx

A given value of y can result from exactly onevalue of x.

x

bx

1

x

logbx

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A Function that is not Single-Valued

Example y = x2

A given value of y can result from one of twovalues of x.

x

y

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Logarithmic Equations

Example Given ln(-x) = ln(x2 6), find x.

Log functions are single-valued,

=

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Exponential Equations

Example Given e5x = e3x+1, find x.

Exponential functions are single-valued,

=

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Problem

Page 192 problem 49.

Find x.

log x + log(10x) = 3

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Logarithmic & Exponential Equations

Principles in Practice 2, p. 187

The sales manager at a fast food chain finds that

breakfast sales begin to fall after the end of a

promotional campaign. The sales in dollars as a

function of the number of days, d, after the

campaigns end are given by

S = 800(4/3)-0.1d.

If the manager does not want sales to drop below

$450 per day before starting a new campaign,

when should he start such a campaign?18

Principles in Practice 2, p. 187S = 800 (4/3)-0.1d where

S is sales in $ per day,

d is no. days after a promotional campaign.

Answer

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Problem

Page 186 Q54.

In statistics the sample regression equation

y = abx is reduced to a linear form by takinglogarithms of bothe sides. Express log y interms of x, log a and log b and explain what ismeant by saying that the resulting expressionis linear?

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Problem

Page 191 Q43

The demand equation for a consumer product isq= 80 2P. Solve for p and express youranswer in terms of common logarithms.Evaluate p to two decimal places when q =60.