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EXPONENTIAL FUNCTIONS PAGE 1 of 16

𝑬 ∙ 𝑑𝑨 =𝑞𝜀(

𝑩 ∙ 𝑑𝑨 = 0

𝑬 ∙ 𝑑𝑺 = −𝑑Φ. 𝑑𝒕

𝑩 ∙ 𝑑𝑺 = 𝜇(𝑖 + 𝜇(𝜀(𝑑Φ3 𝑑𝒕

MATHEMATICAL METHODS UNIT 2 CHAPTER 11 – 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 11 – SET QUESTIONS EXERCISE 11.2: INDICES AS EXPONENTS

2, 4, 6, 8, 10ac, 11ace, 12ac, 13ace, 14ace, 15, 17 EXERCISE 11.3: INDICES AS LOGARITHMS

2, 4, 6, 8, 10ace, 11ace, 12ac, 13ac, 14ace, 15ace, 18ace EXERCISE 11.4: GRAPHS OF EXPONENTIAL FUNCTIONS

1b, 2, 3b, 4, 5b, 6, 7, 9 EXERCISE 11.5: APPLICATIONS OF EXPONENTIAL FUNCTIONS

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14 EXERCISE 11.6: INVERSE OF EXPONENTIAL FUNCTIONS

1a, 2, 5, 6, 14

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Table of Contents

11.2 INDICES AS EXPONENTS 3 INDEX OR EXPONENTIAL FORM 3

REVIEW OF INDEX LAWS 3 FRACTIONAL INDICES 3

Example 1 (Q1) 3 INDICIAL EQUATIONS 4

METHOD OF EQUATING INDICES 4 Example 2 (Q3) 4

INDICIAL EQUATIONS WHICH REDUCE TO QUADRATIC FORM 4 Example 3 (Q5) 4

CAS CALCULATOR 5 SCIENTIFIC NOTATION (STANDARD FORM) 5 SIGNIFICANT FIGURES 5

Worked Example 4 5 11.3 INDICES AS LOGARITHMS 6 INDEX-‐LOGARITHM FORMS 6

USE OF A CALCULATOR 6 Example 5 6

LOG LAWS 7 PROOFS OF THE LOGARITHM LAWS 7

Example 6 8 Example 7 8 Example 8 8 CAS calculator 8 Example 9 9

CONVENTION 9 Example 10 (Q5) 9

EQUATIONS CONTAINING LOGARITHMS 9 Example 11 (Q7) 9

11.4 GRAPHS OF EXPONENTIAL FUNCTIONS 10 EXPONENTIAL FUNCTIONS 10

Example 12 (Q1) 11 TRANSLATIONS OF EXPONENTIAL GRAPHS 11

THE GRAPH OF 𝑦 = 𝑎𝑥

+ 𝑘 11 THE GRAPH OF 𝑦 = 𝑎𝑥 − ℎ 11

Example 13 (Q3) 12 Dilations 12

COMBINATIONS OF TRANSFORMATIONS 12 Example 14 (Q5a) 12

11.5 APPLICATIONS OF EXPONENTIAL FUNCTIONS 13 EXPONENTIAL GROWTH AND DECAY MODELS 13

Example 15 (Q9) 13 ANALYSING DATA 14

Example 16 (Q3) 14 11.6 INVERSE OF EXPONENTIAL FUNCTIONS 15 THE INVERSE OF 𝒚 = 𝒂𝒙, 𝒂 ∈ 𝑹 +\{𝟏} 15

Worked Example 14a 15 RELATIONSHIPS BETWEEN THE INVERSE PAIRS 16

Worked Example 16 16


11.2 Indices as Exponents Index or exponential form

When the number 8 is expressed as a power of 2, it is written as 8 = 2F. 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:

From these, it follows that:

Fractional Indices

Recall: 𝑎GH = 𝑎 and 𝑎

GHI= 𝑎H I

= 𝑎 𝑜𝑟 𝑎GHI= 𝑎

HH = 𝑎L = 𝑎

hence

Example 1 (Q1)

a) Express IGMN×PGQHN

LRGMN as a power of 2

b) Evaluate 27THU + VW

PL

GH


Indicial equations

An indicial equation has the unknown variable as an exponent. In this section we shall consider indicial equations which have rational solutions. Method of equating indices If index laws can be used to express both sides of an equation as single powers of the same base, then this allows indices to be equated.

Example 2 (Q3)

Solve IXYMU×PZMHY

VY= 1 for 𝑥.

Indicial equations which reduce to quadratic form

The technique of substitution to form a quadratic equation may be applicable to indicial equations. To solve equations of the form 𝑝×𝑎2𝑥

+ 𝑞×𝑎𝑥

+ 𝑟 = 0: • Note that 𝑎2𝑥

= (𝑎𝑥)2. • Reduce the indicial equation to quadratic form by using a substitution for 𝑎𝑥. • Solve the quadratic and then substitute back for 𝑎𝑥. • Since 𝑎𝑥 must always be positive, solutions for x can only be obtained for 𝑎𝑥

> 0; reject any negative or zero values for 𝑎𝑥.

Example 3 (Q5) Solve 30×102𝑥

+ 17×10𝑥 − 2 = 0 for 𝑥.


CAS calculator

Solve for 𝑥 given 3TIefL = LVX writing your answer to 2

decimal places. Scientific notation (standard form) Index notation provides a convenient way to express numbers which are either very large or very small.

Writing a number as a × 10b (the product of a number a where1 ≤ a < 10 and a power of 10) is known as writing the number in scientific notation (or standard form). The age of the earth since the Big Bang is estimated to

be4.54 × 109 years, while the mass of a carbon atom is approximately 1.994 × 10−23 grams. These numbers are written in scientific notation.

Significant figures When a number is expressed in scientific notation as either a × 10b or a × 10-‐b, the number of digits in a determines the number of significant figures in the basic numeral. The age of the Earth is 4.54 × 109 years in scientific notation or 4 540 000 000 years to three significant figures. To one significant figure, the age would be 5 000 000 000 years. Worked Example 4

a) Express each of the following numerals in scientific notation and state the number of significant figures each numeral contains. i. 3 266 400

ii. 0.009 876 03

b) Express the following as basic numerals.

i. 4.54 × 109

ii. 1.037 × 10−5


11.3 Indices as logarithms Not all solutions to indicial equations are rational. To obtain the solution to an equation such as 2x = 5, we need to learn about logarithms.

Index-‐logarithm forms

A logarithm is also another name for an index. The index statement, 𝑛 = 𝑎𝑥, with base 𝑎 and index 𝑥, can be expressed with the index as the subject. This is called the logarithm statement and is written as 𝑥 = log𝑎(𝑛). The statement is read as ‘𝑥 equals the log to base 𝑎 of 𝑛’ (adopting the abbreviation of ‘log’ for logarithm).

Use of a calculator

Calculators have two inbuilt logarithmic functions. Base 10 logarithms are obtained from the LOG key. Thus log10(2) is evaluated as log(2), giving the value of 0.3010 to 4 decimal places. Base e logarithms are obtained from the LN key. Thus loge( 2) is evaluated as ln (2), giving the value 0.6931 to 4 decimal places. Base e logarithms are called natural logarithms. When evaluating logarithms, we need to write each equation in exponential form and solve the same way we solve indicial equations. Example 5 Evaluate the following without a calculator.

(a) logR 216

(b) logILP


Log laws There are five log laws:

Note that there is no logarithm law for either the product or quotient of logarithms or for expressions such as loga(m ± n).

Proofs of the logarithm laws 1. Consider the index statement 𝑎0

= 1 ∴ 𝑙𝑜𝑔𝑎(1) = 0 2. Consider the index statement. 𝑎1

= 𝑎 ∴ 𝑙𝑜𝑔𝑎(𝑎) = 1 3. Let 𝑥 = 𝑙𝑜𝑔𝑎(𝑚) 𝑎𝑛𝑑 𝑦 = 𝑙𝑜𝑔𝑎(𝑛)∴ 𝑚 = 𝑎𝑥 and 𝑛 = 𝑎𝑦

𝑚𝑛 = 𝑎𝑥

× 𝑎𝑦

= 𝑎efo Convert to logarithm form: 𝑥 + 𝑦 = 𝑙𝑜𝑔𝑎(𝑚𝑛) Substitute back for 𝑥 and 𝑦: 𝑙𝑜𝑔𝑎(𝑚) + 𝑙𝑜𝑔𝑎(𝑛) = 𝑙𝑜𝑔𝑎(𝑚𝑛)

4. With 𝑥 and 𝑦 as given in law 3: 𝑚𝑛 =

𝑎e

𝑎o = 𝑎eTo

Converting to logarithm form, and then substituting back for 𝑥 and 𝑦 gives:

𝑥 − 𝑦 = logp𝑚𝑛

Substitute back for 𝑥 and 𝑦:

∴ 𝑙𝑜𝑔𝑎(𝑚) − 𝑙𝑜𝑔𝑎(𝑛) = logp𝑚𝑛

5. With x as given in law 3: 𝑥 = 𝑙𝑜𝑔𝑎(𝑚) ∴ 𝑚 = 𝑎𝑥

Raise both sides to the power p: ∴ (𝑚)𝑝

= (𝑎𝑥)𝑝

∴ 𝑚𝑝

= 𝑎𝑝𝑥

Express as a logarithm statement with base a:

𝑝𝑥 = 𝑙𝑜𝑔𝑎(𝑚𝑝)

Substitute back for x: ∴ 𝑝𝑙𝑜𝑔𝑎(𝑚) = 𝑙𝑜𝑔𝑎(𝑚

𝑝) It is important to remember that the log laws only work if the base is the same for each term. To solve logarithmic equations, it is usually best to:

• simplify first using log laws

• express it in exponential form

• solve it as required


Example 6 Simplify each of the following and evaluate where possible, without a calculator.

(a) logL( 5 + logL( 4

(b) logI 12 + logI 8 − logI 3

Example 7 Simplify 3 logI 5 − 2 logI 10. Example 8 Simplify each of the following:

(a) stuv VWstuv FVF

(b) 2 logL( 𝑥 + 1

(c) 5 logL( 𝑥 − 2 CAS calculator Evaluate each of the following expressions, correct to 3 decimal places.

(a) logI 5

(b) logw 8


Example 9

a) Find 𝑥 if logF 9 = 𝑥 − 2.

b) Solve for 𝑥 if logR 𝑥 = −2.

Sometimes when using a calculator which only has bases 10 and 𝑒, the logarithmic equation needs to be rewritten to these bases. This can be done by taking the log of both sides to the same base, then simplifying and solving as required. Convention There is a convention that if the base of a logarithm is not stated, this implies it is base 10. As it is on a calculator, log(n) represents log10(n). When working with base 10 logarithms it can be convenient to adopt this convention. Example 10 (Q5) a) State the exact solution to 7𝑥

= 15 and

calculate its value to 3 decimal places.

b) Calculate the exact value and the value to 3 decimal places the solution to the equation: 3Iefz = 4𝑥.

Equations containing logarithms Example 11 (Q7)

Solve the equation logF(𝑥) + logF(2𝑥 + 1) = 1 for 𝑥.


11.4 Graphs of Exponential functions Exponential functions

Are functions of the form 𝑓: 𝑅 → 𝑅, 𝑓(𝑥) = 𝑎e, 𝑎 ∈ 𝑅f \ 1 . They provide mathematical models of exponential growth and exponential decay situations such as population increase and radioactive decay respectively.

The graph of 𝒚 = 𝒂𝒙 where 𝒂 > 𝟏

The graph of 𝒚 = 𝒂𝒙 where 𝟎 < 𝒂 < 𝟏


Example 12 (Q1) a) On the same set of axes, sketchthe graphs of 𝑦 = 3𝑥 and 𝑦 = −3𝑥, stating their ranges.

b) Give a possible equation for the graph shown.

Translations of exponential graphs Once the basic exponential growth or exponential decay shapes are known, the graphs of exponential functions can be translated in similar ways to graphs of any other functions previously studied. The graph of 𝑦 = 𝑎e

+ 𝑘 Under a vertical translation the position of the asymptote will be altered to 𝑦 = 𝑘. If 𝑘 < 0, the graph will have 𝑥-‐axis intercepts which are found by solving the exponential equation 𝑎𝑥 + 𝑘 = 0. The graph of 𝑦 = 𝑎eT� Under a horizontal translation, the asymptote is unaffected. The point on the y-‐axis will no longer occur at 𝑦 = 1. An additional point to the y-‐intercept that can be helpful to locate is the one where 𝑥 = ℎ, since 𝑎eT� will equal 1 when 𝑥 = ℎ.


Example 13 (Q3) Sketch the graph of the following and state the range.

𝑦 = 4𝑥

− 2

Dilations Exponential functions of the form 𝑦 = 𝑏×𝑎𝑥 have been dilated by a factor 𝑏 (𝑏 > 0) from the 𝑥-‐axis. This affects the y-‐intercept, but the asymptote remains at y = 0. Exponential functions of the form 𝑦 = 𝑎�e have been dilated by a factor L

� (𝑛 > 0) from the 𝑦-‐axis. This affects the

steepness of the graph but does not affect either the 𝑦-‐intercept or the asymptote.

Combinations of transformations Exponential functions with equations 𝑦 = 𝑏 × 𝑎�(eT�) + 𝑘 are derived from the basic graph of 𝑦 = 𝑎𝑥 by applying a combination of transformations. The key features to identify, to sketch the graphs of such exponential functions are:

• the asymptote • the 𝑦-‐intercept • the 𝑥-‐intercept, if there is one.

Another point that can be obtained simply could provide assurance about the shape. Always aim to show at least two points on the graph.

Example 14 (Q5a) Sketch the graph the following and state the range.


11.5 Applications of exponential functions The importance of exponential functions lies in models of phenomena involving growth and decay situations, in chemical and physical laws of nature and in higher-‐level mathematical analysis. Exponential growth and decay models

For time 𝑡, the exponential function defined by 𝑦 = 𝑏 × 𝑎𝑛𝑡 where 𝑎 > 1 represents:

• exponential growth over time if 𝑛 > 0 • exponential decay over time if𝑛 < 0 The domain of this function would be restricted according to the waythe independent time variable 𝑡 is defined.

In some mathematical models such as population growth, the initial population may be represented by a symbol such as 𝑁0. For the radioactive exponential decay model, the time it takes for 50% of the initial amount of the substance to decay is called its half-‐life.

Example 15 (Q9) The contents of a meat pie immediately after being heated in a microwave have a temperature of 95 °𝐶. The pie is removed from the microwave and left to cool.A model for the temperature of the pie as it cools is given by 𝑇 = 𝑎×3T(.LF� + 25 where 𝑇 is the temperature after 𝑡 minutes of cooling. a) Calculate the value of 𝑎.

b) What is the temperature of the contents of the pie after being left to cool for 2 minutes?

c) Determine how long, to the nearest minute, it will take for the contents of the meat pie to cool to 65 °C.


d) Sketch the graph showing the temperature over time and state the temperature to which this model predicts the contents of the pie will eventually cool if left unattended.

Analysing data

One method for detecting if data has an exponential relationship can be carried out using logarithms. If the data is suspected of following an exponential rule such as 𝑦 = 𝐴×10𝑘𝑥, then the graph of log(𝑦) against 𝑥 should be linear.

This equation can be written in the form: 𝑌 = 𝑘𝑥 + 𝑐 where 𝑌 = log(𝑦) and 𝑐 = log(𝐴) The graph of 𝑌 𝑣𝑒𝑟𝑠𝑢𝑠 𝑥 is a straight line with gradient 𝑘 and vertical axis 𝑌-‐intercept (0, log(𝐴)). Such an analysis is called a semi-‐log plot. While experimental data is unlikely to give a perfect fit, the equation would describe the line of best fit for the data. Logarithms can also be effective in determining a power law that connects variables. If the law connecting the variables is of the form 𝑦 = 𝑥� then log(𝑦) = 𝑝log(𝑥). Plotting log(𝑦) values against log(𝑥) values will give a straight line of gradient 𝑝 if the data does follow such a law. Such an analysis is called a log-‐log plot. Example 16 (Q3) For a set of data (𝑥, 𝑦), plotting log(𝑦) versus log(𝑥) gave the straight line shown in the diagram. From the equation of the graph and hence determine the rule connecting 𝑦 and 𝑥.


11.6 Inverse of exponential functions The inverse of 𝒚 = 𝒂𝒙, 𝒂 ∈ 𝑹f\{𝟏}

The exponential function has a one-‐to-‐one correspondence so its inverse must also be a function. To form the inverse of 𝑦 = 𝑎𝑥, interchange the 𝑥-‐ and 𝑦-‐coordinates.

function: 𝑦 = 𝑎𝑥 domain 𝑅, range 𝑅+

inverse function: 𝑥 = 𝑎𝑦 domain 𝑅+, range 𝑅

∴ 𝑦 = log𝑎(𝑥)

Therefore, the inverse of an exponential function is a logarithmic function: 𝑦 = log𝑎(𝑥) and 𝑦 = 𝑎𝑥 are the rules for a pair of inverse functions. This means the graph of 𝑦 = log𝑎(𝑥) can be obtained by reflecting the graph of 𝑦 = 𝑎𝑥 in the line 𝑦 = 𝑥.

Worked Example 14a Form the exponential rule for the inverse of y = log2(x) and hence deduce the graph of y = log2(x) from the graph of the exponential.


Relationships between the inverse pairs

As the exponential and logarithmic functions are a pair of inverses, each ‘undoes’ the effect of the other. From this it follows that:

The first of these statements could also be explained using logarithm laws:

log𝑎(𝑎𝑥) = 𝑥 log𝑎(𝑎)

= 𝑥×1 = 𝑥

The second statement can also be explained from the index-‐logarithm definition that 𝑎𝑛 = 𝑥 ⇔ 𝑛 = log𝑎(𝑥). Replacing 𝑛 by its logarithm form in the definition gives:

𝑎𝑛

= 𝑥 𝑎stu�(e)

= 𝑥 Worked Example 16

a) Simplify log12(22𝑥×3𝑥) using the inverse relationship between exponentials and logarithms.

b) Evaluate 10I stuG�(z)

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