ATHEMATICAL!METHODS!UNIT 2 C 13 EXPONENTIAL!FUNCTIONS › uploads › 5 › 2 › 6 › 4 ›...

EXPONENTIAL FUNCTIONS PAGE 1 of 20

!𝑬 ∙ 𝑑𝑨 =𝑞𝜀)

!𝑩 ∙ 𝑑𝑨 = 0

!𝑬 ∙ 𝑑𝑺 = −𝑑Φ/ 𝑑𝒕⁄

!𝑩 ∙ 𝑑𝑺 = 𝜇)𝑖 + 𝜇)𝜀)𝑑Φ5 𝑑𝒕⁄

MATHEMATICAL METHODS UNIT 2 CHAPTER 13 – EXPONENTIAL FUNCTIONS Key knowledge • the key features and properties of power and polynomial functions and their graphs • the effect of transformations of the plane, dilation, reflection in axes, translation and simple combinations of these transformations, on the graphs of linear and power functions

• the relation between the graph of a one-‐to-‐one function, its inverse function and reflection in the line y = x • key mathematical content from one or more areas of study related to a given context • specific and general formulations of concepts used to derive results for analysis within a given context • the role of examples, counter-‐examples and general cases in working mathematically • inferences from analysis and their use to draw valid conclusions related to a given context. Key skills • draw graphs of polynomial functions of low degree, simple power functions and simple relations that are not functions • specify the relevance of key mathematical content from one or more areas of study to the investigation of various questions in a given context

• develop mathematical formulations of specific and general cases used to derive results for analysis within a given context • use a variety of techniques to verify results • make inferences from analysis and use these to draw valid conclusions related to a given context • communicate conclusions using both mathematical expression and everyday language, in particular, the interpretation of mathematics with respect to the context.

CHAPTER 13 – SET QUESTIONS EXERCISE 13A: THE INDEX LAWS, 2ac, 3ac, 4ac, 5ac, 6a, 7ac, 8ac, 9aceg, 10ac, 11ace, 12ac,

13ac EXERCISE 13B: RATIONAL INDICES 1acegikm, 2ace, 3ace EXERCISE 13C: GRAPHS OF EXPONENTIAL FUNCTIONS 1ac, 2ac, 3, 4, 5ace, 6 EXERCISE 13D: SOLVING EXPONENTIAL EQUATIONS & INEQUALITIES 1acegi, 2acegikm, 3ac, 4ac, 5ac, 6aceg

EXERCISE 13E: LOGARITHMS 1ac, 2aceg, 3acegik, 4aceg, 5acegi, 6ace, 7, 8, 9, 10aceg EXERCISE 13F: USING LOGARITHMS TO SOLVE EXPONENTIAL EQUATIONS AND INEQUALITIES 1ace, 2ac, 3ace, 4ace

EXERCISE 13G: GRAPHS OF LOGARITHM FUNCTIONS 1ace, 2ac, 3aceg, 4ace, 5, 6, 8 EXERCISE 13H: EXPONENTIAL MODELS AND APPLICATIONS 1, 2, 3, 4, 6, 8, 9, 11, 16

MORE RESOURCES http://drweiser.weebly.com


Table of Contents

13A THE INDEX LAWS 3 REVIEW OF INDEX LAWS 3

Example: Multiplying powers 3 Example: Dividing powers Q2(b) 3 Example: negative integer indices Q4b and Raising the power Q5b 4 Example: Products and quotients Q6b 4

WORKING WITH A NEGATIVE BASE 4 Example 6 Negative Bases 4

USING PRIME DECOMPOSITION 4 Example 7 Prime Decomposition 4

13B RATIONAL INDICES 6 Example 8: Rational Indices 6 Example 9 6

13C GRAPHS OF EXPONENTIAL FUNCTIONS 7 GRAPH OF Y=AX WHEN A >1 7 GRAPH OF Y=AX WHEN 0<A<1 7 GRAPHS OF Y = AX IN GENERAL 7 TRANSFORMATIONS OF EXPONENTIAL GRAPHS 8

13D SOLVING EXPONENTIAL EQUATIONS AND INEQUALITIES 10 SOLUTION OF EQUATIONS 10

Example solve for 𝑥 10 Example 17 11

SOLUTION OF INEQUALITIES 11 Example Inequalities 11

13E LOGARITHMS 12 Example Q1(b) and (d) 12

LAWS OF LOGARITHMS 12 Example Logarithm Laws Q4(b) and (h) 13 Example 22 Solve for 𝑥 13

13F USING LOGARITHMS TO SOLVE EXPONENTIAL EQUATIONS AND INEQUALITIES 14 Example Q1(b) and (f) 14 Example 24 14 Example 25 14

EXPONENTIAL GRAPHS REVISITED 14 13G GRAPHS OF LOGARITHM FUNCTIONS 15 GRAPH OF Y=LOGAX WHEN A>1 15

Y = LOG2 X 15 Y = LOG10 X 15 GRAPH OF Y=LOGAX WHEN 0<A<1 16 INVERSES 16 TRANSFORMATIONS OF LOGARITHM GRAPHS 16

13H EXPONENTIAL MODELS AND APPLICATIONS 18 EXPONENTIAL GROWTH AND DECAY 18

CELL GROWTH 18 RADIOACTIVE DECAY 18 POPULATION GROWTH 19 DETERMINING EXPONENTIAL RULES 19


13A The Index Laws When the number 8 is expressed as a power of 2, it is written as 8 = 2:. In this form, the base is 2 and the power (also known as index or exponent) is 3. Review of index laws Recall the basic index laws:

Example: Multiplying powers

a) Simplify 𝑥;𝑦: × 𝑥>𝑦

b) Simplify 2? × 2;?@;

Example: Dividing powers Q2(b)

Simplify ABC×ADCEF

AGC


Example: negative integer indices Q4b and Raising the power Q5b

Evaluate H14K−3

Simplify MHN:KO;P:

Example: Products and quotients Q6b

Simplify HQRGAG

RADSDK:÷ (𝑎;𝑏ON𝑐):

Working with a negative base The power (−a)n can be written as (−1 × a)n = (−1)n(a)n. We note that:

• If n is even then (−1)n =1. • If n is odd then (−1)n =−1.

Hence, if a is a positive number, then the number (−a)n is positive when n is even and negative when n is odd. Example 6 Negative Bases1 Simplify each of the following

a) (−3)O:

b) (−5𝑎): c) (−2𝑎): × 3𝑎;

Using prime decomposition Bases that are composite numbers are often best factored into primes before further calculations are undertaken.

Example 7 Prime Decomposition2 Simplify the following, expressing the answers in positive-index form:

a) 12[ × 18O;[

b) :\G×]^×N;\G

_\^×;\D c)

:D`×]`

a`×:`

1 https://seniormaths.cambridge.edu.au/lessonSection/lesson.action#/resources/video/78015/ 2 https://seniormaths.cambridge.edu.au/lessonSection/lesson.action#/resources/video/78015/


13B Rational Indices

Example 8: Rational Indices Evaluate the following

a) (−64)FG

b) 9OFD c) 64O

DG

Example 93

a) :F^×√]× √;^

N]G^

b) (𝑥O;𝑦)FD × H ?

e\GK>

3 https://seniormaths.cambridge.edu.au/lessonSection/lesson.action#/resources/video/78023/


13C Graphs of exponential functions Two types of graphs of exponential functions will be examined. Graph of y=ax when a >1 We can make the following observations about graphs of the form y = ax where a > 1: • The x-axis is said to be an asymptote. As x → −∞, y → 0+.

This is read: As x approaches negative infinity, y approaches 0 from the positive side.

• As the x-values increase, the y-values increase. • The y-axis intercept is at (0,1). • The range of the function is R+.

Graph of y=ax when 0<a<1 We can make the following observations about graphs of the form y = ax where 0 < a < 1: • The x-axis is an asymptote. As the x-values increase, the graph approaches the x-axis from above. This is

written: As x → ∞, y → 0+. • The y-axis intercept is at (0,1). • The range of the function is R+.

Graphs of y = ax in general In both cases a > 1 and 0 < a < 1, we can write 𝑦 = 𝑎𝑥 as 𝑦 = 𝑏O?, where 𝑏 = N

R.

The graph of 𝑦 = 𝑏O?is obtained from the graph of 𝑦 = 𝑏? by a reflection in the y-axis. Thus, for example, the

graph of 𝑦 = HN;K?

is obtained from the graph of y = 2x by a reflection in the y-axis, and vice versa.


Transformations of exponential graphs The techniques for transformations that were introduced in earlier chapters are now applied to the graphs of exponential functions.


13D Solving exponential equations and inequalities Solution of equations One method without using a calculator is to express both sides of the equation as powers with the same base and then equate the indices (since ax = ay implies x = y, for any a ∈ R+ \ {1}). Example solve for 𝑥 Q1(b): 4? = 64

Q1(h): 4O? = N]>

Q2(b): 3;[O> = 1

Q2(k): 25[@N = 5 × 390625

Sometimes solving an exponential equation involves solving a polynomial equation first. In the following example, the solution of a quadratic equation is necessary.


Example 17 Solve:

a) 9x =12×3x −27

b) 32x = 27 – 6 × 3x

Solution of inequalities The following two properties are useful when solving inequalities:

§ 𝑎𝑥 > 𝑎𝑦 ⇔ 𝑥 > 𝑦, when 𝑎 ∈ (1,∞) § 𝑎𝑥 > 𝑎𝑦

⇔ 𝑥 < 𝑦, when 𝑎 ∈ (0,1)

Example Inequalities Solve for 𝑥 in each of the following:

a) 16? > 2

b) 2O:?@N < NN]


13E Logarithms Consider the statement 23 = 8, This may be written in an alternative form: log2 8 = 3, which is read as ‘the logarithm of 8 to the base 2 is equal to 3’.

Note: Since ax is positive, the expression loga y is only defined when y is positive. The logarithm function with base a is the inverse of the exponential function with base a. We will discuss this in Section 13G. Further examples: § 32 = 9 is equivalent to 𝑙𝑜𝑔39 = 2 § 104 = 10 000 is equivalent to 𝑙𝑜𝑔10 10 000 = 4 § 𝑎0 = 1 is equivalent to 𝑙𝑜𝑔𝑎1 = 0 Example Q1(b) and (d)

Without using a calculator, evaluate the following: a) log: 81

b) logN) 0.1

Laws of logarithms The index laws are used to establish rules for computations with logarithms.


Example Logarithm Laws Q4(b) and (h)

Without using a calculator, simplify each of the following: a) log; 16+log; 8

b) :;logR 𝑎 − logR √𝑎

Example 22 Solve for 𝑥4

(a) logQ 𝑥 = 3

(b) logQ(2𝑥 + 1) = 2

(c) log;(2𝑥 + 1) − log;(𝑥 + 1) = 4

(d) log:(𝑥 − 1) + log:(𝑥 + 1) = 1

4 https://seniormaths.cambridge.edu.au/lessonSection/lesson.action#/resources/video/78065/


13F Using logarithms to solve exponential equations and inequalities If a ∈ R+ \ {1}, then the statements ax = b and loga b = x are equivalent. This defining property of logarithms may be used in the solution of exponential equations. Example Q1(b) and (f) Solve using a calculator (b) 2x=0.4

(f) 0.3x=2

Example 24 Solve 32x−1 = 28.

Example 25

Solve the inequality 0.7𝑥 ≥ 0.3

Exponential graphs revisited In Section 13C we graphed exponential functions, but often we could not find the x-axis intercept. Now that we have defined logarithms this can be done.


13G Graphs of logarithm functions Graph of y=logax when a>1

y = log2 x

y = log10 x

Notes: § The point (1,2) is on the graph of y=2x and

the point (2,1) is on the graph of y=log2 x.

§ The point(2,4) is on the graph of y=2x and the point (4,2) is on the graph of y=log2 x.

§ The graph of y=log2 x is the reflection of the

graph of y=2x in the line y=x.

The graph of y = log10 x is the reflection in the line y = x of the graph of y = 10x.

Properties of y=ax, a>1

• domain = R • range = R+ • a0=1 • As x→−∞,y→0+ • y = 0 is an asymptote

Properties of y=loga x, a>1 • domain = R+

• range = R • loga 1 = 0 • as x → 0+, y → −∞ • x = 0 is an asymptote


Graph of y=logax when 0<a<1

Consider 𝑦 = logFD𝑥

logN;𝑥 = 𝑦 ⟺ M

12P

e

= 𝑥

⟺ 2Oe = 𝑥 ⟺ log; 𝑥 = −𝑦 ⟺ 𝑦 = − log; 𝑥

So we have logFD𝑥 = − log; 𝑥.

Inverses

The inverse of a one-to-one function was introduced in Section 5G. • The inverse of 𝑓: 𝑅 → 𝑅, 𝑓(𝑥) = 2𝑥 is the function 𝑓ON: 𝑅@ → 𝑅, 𝑓ON(𝑥) = 𝑙𝑜𝑔2𝑥 • The inverse of 𝑓: 𝑅 → 𝑅, 𝑓(𝑥) = 10𝑥 is the function 𝑓ON: 𝑅@ → 𝑅, 𝑓ON(𝑥) = 𝑙𝑜𝑔10𝑥

Transformations of logarithm graphs Transformations can be applied to the graphs of logarithm functions. This is shown in the following example.


13H Exponential models and applications Exponential growth and decay In the following, we consider a variable A that is subject to exponential change. Let A be the quantity at time t. Then A = A0bt, where A0 is the initial quantity and b is a positive constant.

If b > 1, the model represents growth: • Growth of cells • Population growth

• Continuously compounded interest

If b < 1, the model represents decay: • Radioactive decay

• Cooling of materials

Cell growth Suppose a particular type of bacteria cell divides into two new cells every TD minutes. Let N0 be the initial number of cells of this type. Then after t minutes the number of cells, N, is given by the formula

𝑁 = 𝑁) × 2{|}

where TD is called the generation time. Here we are only dealing with the type of reproduction where the cell divides in two. For most known bacteria that can be cultured, generation times range from about 15 minutes to 1 hour.

Radioactive decay Radioactive materials decay so that the amount of radioactive material, A, present at time t (in years) is given by the formula

𝐴 = 𝐴0 × 2O�{ where A0 is the initial amount and k is a positive constant that depends on the type of material. A radioactive substance is often described in terms of its half-life, which is the time required for half the material to decay.


Population growth It is sometimes possible to model population growth through exponential models.

Determining exponential rules We have looked at determining rules for functions in Chapters 2 to 6. We look at one very useful case for exponential functions.


In many practical situations, the relationship between variables is exponential.

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