Chapter 14 Exponentials And Logarithms - Year 1 Flashcards
What are the 3 special log cases?
Give an example of an exponential?
2^x
Give an example of a polynomial?
x^2
What do you know about the graph 2^x including the general shape?
It is an exponential
It crosses the y axis at (0,1)
It has an asymptote at y=0
How does the base effect of an exponential effect it?
(When the base is greater than 1)
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
Draw the graph of y=1^x
How does the base effect of an exponential effect it?
(When the base is less than 1) (but greater than 0)
It will be mirrored in the y axis
What is the difference between a^x and (1/a)^x
y^x is said to be exponentially growing
(1/y)^x is said to be exponentially decaying
Prove that (1/a)^x is a reflection in the y axis of a^x
If y=2^x how do you draw y=2^x+a?
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)
How do you differentiate e?
What is special about e?
Its gradient function is the same as its normal function
What is Eulers number?
e
2.71
How does a power effect a graph with e as its base?
Urbdnddj
How do you transform a graph of y= a+e^bx
Include all possible values/signs of a and b.
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:
If you are given an exponential how do you know the initial population?
Make t=0
The answer should be the constant without e
How do you find the initial rate of population growth when you have an exponential?
What if it wasn’t initial
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
t
Interpret the significance of the sign on a differentiated exponential equation.
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
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)
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
Write a^x=n as a logarithm
How do you say a logorithm?
Log base a of n
What are the 3 log buttons on the calculator?
Explain:
The main log button on your calculator
Explain:
Log base 10 bu defufult
(This is like how 2sqr() is just sqr
Explain:
The natural log. This means log base e.
It is the inverse of e^x
What are the 3 main laws of logs?
What are you commonly asked to do with logs?
Convert 2 into 1
Draw and label anything intresting
What are 2 common mistakes with logs
How do you solve an equation when you have an unknown in the logorithm?
You convert it into the form a^b=c and solve from there
What do you do when you have 2 logs equal to each other in an equation?
Bring them both onto the same side and convert them into 1 log
How do you solve logarithmic simultaneous equations?
Convert the logs into the form a^b=c form
Then solve the simultaneous equations
What trick may be used to narrow down the options in a simultaneous equation? (2)
x has to be positive
x>y so if you get x=4 and y=6 this can’t be the answer
If it tells you to only plot 1 graph what doesn’t matter?
The magnitude (relative to other graphs. Only the general shape matters)
Sketch 2^-x
Sketch 2^x
Sketch -2^x
Sketch -2^-x
What should you avoid having in logs?
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.
How do you solve an exponential equation?
Take the log of both sides and simplify
Then put it into the calculator
How do you solve an exponential equation when there are 2 exponentials which have different bases?
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)
What is important to remember about log rules?
They work both ways:
What is the inverse of:
x
x
How do you get rid of the natural log in an equation?
Make everything in the equation a power of e
How do you get rid of e^x in an equation?
Take the natural log of everything
How so you solve an equation that has e^x in it?
Take the natural log of both sides
How do you solve an equation that has a natural log in it?
Put everything as a power of e
How should you write In(x) / 2
What is a common natural logarithm mistake?
What does this equal?
When you have a logarithm + or - a number, how should you write it?
At the front
What question may you get that is covered in a different chapter?
Hidden quadratics
How do you solve an exponential hidden quadratic equation when you have e^x and e^-1?
You X everything by e because e^-1 = 1/e^x
Expand:
How can we turn a polynomial into a linear equation?
Taking logs of both sides
log(y) = log(a) - nlog(X)
You can now compare this to y=mX+c
How can we turn an exponential into a linear equation?
Taking logs of both sides
log(y)=log(a)+Xlog(b)
We can compare this to y=mX+c
What does taking logs of an exponential or polynomial equation do to the graph?
Makes it into a straight line
Chapter 14 exponentials
How do you solve an equation with an e to the power of a polynomial?
You take the natural of both sides and rewrite the e term as 2 natural logarithms and then solve from there
Draw the linear graph of y=ax^n
Draw the linear graph of y=ab^x
