Exponential and logarithmic functions

Exponentialand logarithmic

functionsYr 11 maths methods

To define and understand exponential functions. To sketch graphs of the various types of exponential functions. To understand the rules for manipulating exponential and

logarithmic expressions. To solve exponential equations. To evaluate logarithmic expressions. To solve equations using logarithmic methods. To sketch graphs of functions of the form y = logax and simple

transformations of this. To understand and use a range of exponential models. To sketch graphs of exponential functions. To apply exponential functions to solving problems.

Objectives for Term 2

Introduction Functions in which the independent

variable is an index number are called indicial or exponential functions. For example:

f (x) = ax where a > 0 and a ≠ 1 quantities which increase or decrease by a

constant percentage in a particular time can be modelled by an exponential function.

Exponential functions can be seen in everyday life for example in science and medicine (decay of radioactive material, or growth of bacteria like those shown in the photo), and finance ( compound interest and reducing balance loans).

Index laws

Multiplication

am × an = am + n When multiplying two

numbers in index form with the same base, add the indices.

For example, 23 × 24 = (2 × 2 × 2) × (2 × 2 × 2 × 2) = 27

Division

am ÷ an = am - n When dividing two numbers in index form with the same base, subtract the indices.

Raising to a power

(am)n = am × n = amn To raise an indicial expression to a power, multiply the indices.

Raising to the power of zero

a0 = 1, a ≠ 0 Any number raised to the power of zero is equal to one.

Products and quotients

Remember

Questions

Answers (a)

Answers (b)

Answers (c)

Answers (d)

Page 220 Questions 1 – 3

Homework

More Questions

Answer without using your Cauculators

Answer with your calculators

Questions

Answer (a)

Answer (b)

Question

Answer

Page 220 – 221 - Questions 4 – 10

Homework

negative and rational powers

negative powers

Examples

Answer A

Answer B

Rational powers

Examples

Examples

Indicial equations

Indicial equations

Examples

Answer A

Answer B

Answer C

Solve the following

Answer

Answer

Graphs of exponential functions

Graphs of exponential functions

The effect of changing the “a” coeff

-4 -3 -2 -1 0 1 2 3 40

5

10

15

20

25

30

y=2^x y=3^x y=2^-x y=3^-x

The effect of changing the “a” coeff

-4 -3 -2 -1 0 1 2 3 40

5

10

15

20

25

30

y=2^xy=3^x

The effect of changing the “a” coeff

-4 -3 -2 -1 0 1 2 3 40

5

10

15

20

25

30

y=2^-xy=3^-x

Reflections of exponential functions

Reflections of exponential functions

-4 -3 -2 -1 0 1 2 3 40

1

2

3

4

5

6

7

8

9

y=2^xy=2^-x

Reflections of exponential functions

-4 -3 -2 -1 0 1 2 3 4

-10

-8

-6

-4

-2

0

2

4

6

8

10

y=2^xy=-2^x

Horizontal translations of exponential functions

Vertical translations of exponentialfunctions

Dilation from the x-axis

Dilation from the y-axis

Examples

Examples

Calculator time.