Chapter 14 Exponentials And Logarithms - Year 1

Question 1

What are the 3 special log cases?

Answer 1

...when...

Question 2

Give an example of an exponential?

Answer 2

2^x

Question 3

Give an example of a polynomial?

Answer 3

x^2

Question 4

What do you know about the graph 2^x including the general shape?

Answer 4

It is an exponential
It crosses the y axis at (0,1)
It has an asymptote at y=0

Question 5

How does the base effect of an exponential effect it?
(When the base is greater than 1)

Answer 5

As the base gets smaller the smaller base is above the graph of 2^x until they intercect at 0,1 where the smaller base is below the graphs of 2^x
As the base gets larger the larger base is below the graph of of 2^x until they intercect at 0,1 where the larger base is above the graph of 2^x

Question 6

Draw the graph of y=1^x

Answer 6

Question 7

How does the base effect of an exponential effect it?
(When the base is less than 1) (but greater than 0)

Answer 7

It will be mirrored in the y axis

Question 8

What is the difference between a^x and (1/a)^x

Answer 8

y^x is said to be exponentially growing
(1/y)^x is said to be exponentially decaying

Question 9

Prove that (1/a)^x is a reflection in the y axis of a^x

Answer 9

Question 10

If y=2^x how do you draw y=2^x+a?

Answer 10

As long as a is positive you would shift y=2^x to the left for the value of a

If a is negative then you shift to the right
(Because if it is inside the brackets/effecting the x axis you do the opposite of what is expected)

Question 11

How do you differentiate e?

Answer 11

Question 12

What is special about e?

Answer 12

Its gradient function is the same as its normal function

Question 13

What is Eulers number?

Answer 13

e

2.71

Question 14

How does a power effect a graph with e as its base?

Answer 14

Urbdnddj

Question 15

How do you transform a graph of y= a+e^bx
Include all possible values/signs of a and b.

Answer 15

If a is positive you will shift the graph up by a. The same will happen to the asymptote
If a is negative you will shift the graph down by a. The same will happen to the asymptote

If b is negative you will reflect the graph in the y axis
As b gets larger:

Question 16

If you are given an exponential how do you know the initial population?

Answer 16

Make t=0
The answer should be the constant without e

Question 17

How do you find the initial rate of population growth when you have an exponential?

What if it wasn’t initial

Answer 17

Take the derivative
Make sure to use the letters they give you eg dp/dt
Then because it says initial make t=0 and substitute it into the differentiated equation

If it wasn’t initial and said at y years you would substitute t for the value of y

Answer 18

t

Question 19

Interpret the significance of the sign on a differentiated exponential equation.

Answer 19

If it is negative the rate is negative thus the density of the particle is decreasing

If it is positive the rate is positive thus the density of the particle is increasing

Question 20

What do you do if you are asked to sketch the graph of P against t at the end of a question?
(Specific to C14)

Answer 20

Sketch the original function

It looks like a normal exponential
Just look at the power to see if it is positive or negative
The y intercept will be the constant

if t is time then t will be on the x axis

160e^-0.006t

Question 21

Write a^x=n as a logarithm

Answer 21

Question 22

How do you say a logorithm?

Answer 22

Log base a of n

Question 23

What are the 3 log buttons on the calculator?

Answer 23

Question 24

Explain:

Answer 24

The main log button on your calculator

Question 25

Explain:

Answer 25

Log base 10 bu defufult (This is like how 2sqr() is just sqr

Question 26

Explain:

Answer 26

The natural log. This means log base e. It is the inverse of e^x

Question 27

What are the 3 main laws of logs?

Answer 27

Question 28

What are you commonly asked to do with logs?

Answer 28

Convert 2 into 1

Question 29

Draw and label anything intresting

Answer 29

Question 30

What are 2 common mistakes with logs

Answer 30

Question 31

How do you solve an equation when you have an unknown in the logorithm?

Answer 31

You convert it into the form a^b=c and solve from there

Question 32

What do you do when you have 2 logs equal to each other in an equation?

Answer 32

Bring them both onto the same side and convert them into 1 log

Question 33

How do you solve logarithmic simultaneous equations?

Answer 33

Convert the logs into the form a^b=c form Then solve the simultaneous equations

Question 34

What trick may be used to narrow down the options in a simultaneous equation? (2)

Answer 34

x has to be positive x>y so if you get x=4 and y=6 this can't be the answer

Question 35

If it tells you to only plot 1 graph what doesn't matter?

Answer 35

The magnitude (relative to other graphs. Only the general shape matters)

Question 36

Sketch 2^-x

Answer 36

Question 37

Sketch 2^x

Answer 37

Question 38

Sketch -2^x

Answer 38

Question 39

Sketch -2^-x

Answer 39

Question 40

What should you avoid having in logs?

Answer 40

A fraction. You should raise the number to a negative index to get the number in the log to a number. Then you can bring the negative power to the front of the log.

Question 41

How do you solve an exponential equation?

Answer 41

Take the log of both sides and simplify Then put it into the calculator

Question 42

How do you solve an exponential equation when there are 2 exponentials which have different bases?

Answer 42

You take the log of both sudes but you don't need to specify the base. Then simplify and solve When solving you can just assume that the base is log10(x)

Question 43

What is important to remember about log rules?

Answer 43

They work both ways:

Question 44

What is the inverse of:

Answer 44

Question 45

Answer 45

Question 46

Answer 46

x

Question 47

Answer 47

x

Question 48

How do you get rid of the natural log in an equation?

Answer 48

Make everything in the equation a power of e

Question 49

How do you get rid of e^x in an equation?

Answer 49

Take the natural log of everything

Question 50

How so you solve an equation that has e^x in it?

Answer 50

Take the natural log of both sides

Question 51

How do you solve an equation that has a natural log in it?

Answer 51

Put everything as a power of e

Question 52

How should you write In(x) / 2

Answer 52

Question 53

What is a common natural logarithm mistake?

Answer 53

Question 54

What does this equal?

Answer 54

Question 55

When you have a logarithm + or - a number, how should you write it?

Answer 55

At the front

Question 56

What question may you get that is covered in a different chapter?

Answer 56

Hidden quadratics

Question 57

How do you solve an exponential hidden quadratic equation when you have e^x and e^-1?

Answer 57

You X everything by e because e^-1 = 1/e^x

Question 58

Expand:

Answer 58

Question 59

How can we turn a polynomial into a linear equation?

Answer 59

Taking logs of both sides log(y) = log(a) - nlog(X) You can now compare this to y=mX+c

Question 60

How can we turn an exponential into a linear equation?

Answer 60

Taking logs of both sides log(y)=log(a)+Xlog(b) We can compare this to y=mX+c

Question 61

What does taking logs of an exponential or polynomial equation do to the graph?

Answer 61

Makes it into a straight line

Question 62

Chapter 14 exponentials

Question 63

How do you solve an equation with an e to the power of a polynomial?

Answer 63

You take the natural of both sides and rewrite the e term as 2 natural logarithms and then solve from there

Question 64

Draw the linear graph of y=ax^n

Answer 64

Question 65

Draw the linear graph of y=ab^x