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Maths Prelim > Logarithms > Flashcards

Logarithms Flashcards

1
Q

What is a logarithm

A

related to exponentials, used to solve exponential equations

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2
Q

2^3 = 8 =…

A

log2 8 = 3

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3
Q

Restriction on log function

A

base a>1

base a cannot = 0

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4
Q

if a^x = y…

A

loga y = x

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5
Q

Graph of y=a^x characteristics

A 
a^x always positive,
y-int: (0,1)
Domain: (-8, 8)
Range: (0, 8)
Asymptote: y=0
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6
Q

Graph of y=a^-x

A

Reflection of y=a^x about the y-axis

same y-int, domain, range, asymptote

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7
Q

When a = 1/2…

A

= 2^-x

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8
Q

Graph of y=loga x characteristics

A

x-int: (1,0)

domain: (0,8)
range: (-8, 8)
asymptote: x=0

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9
Q

loga (mn)=…

A

loga m + loga n

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10
Q

loga (m/n)=…

A

loga m - loga n

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11
Q

loga (m^n)=…

A

n loga m

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12
Q

loga 1=…

A

0

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13
Q

loga a=…

A

1

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14
Q

loga a^n=…

A

n

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15
Q

a^loga x=…

A

x

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16
Q

change of base law

A

loga x = logb x/logb a

x on top, base on bottom

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17
Q

How to find derivative of exponential function using first principles

A

same way as usual (lim h–> 0 f(x+h) -f(x)/h)

but substitute a small value of x into h instead of 0 and put x value in front

f(x) = a^x
f'(x) = a^x lim h->0 a^h -1/h
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18
Q

what is Euler’s number

A

e^x

the number that gives exactly the same graph for a derivative function

19
Q

derivative of e^x

A

e^x

20
Q

derivative of ke^x

A

ke^x

21
Q

How to differentiate y=ke^f(x)

A 
Use chain rule (dy/dx = dy/du x du/dx)
(let u = f(x))
22
Q

How to differentiate y=ke^f(x) fast way

A

Bring down derivative of power and multiply

power stays the same

23
Q

what is the natural logarithm

A 
logarithm to base e
loge x (ln x)
24
Q

y=e^x and y=loge x are…

A

symmetrical about the line y=x

25
Q

for inverse functions

A

the x and y values are interchanged

26
Q

Why do you always have to check for valid solutions when solving log equations

A

because there is restrictions

a > 0, a cannot = 1, and x > 1

27
Q

Why do we need logs to solve equations

A

Previously we solved equations by changing the base to the same and equating indicies (2^x+1 = 16
2^x+1 = 2^4
x+1=4, x=3)
but we can only do that when the base can be changed, so we use logs to solve these equations

28
Q

How to solve equations with logs

A
  1. Rewrite in log form and use change of base rule
  2. Take the log of both sides, then use log laws
    (don’t forget to check if a log is negative when working with inequalities)
29
Q

Natural log laws

A

loge e^x = x

e^loge e^x = x

30
Q

If both sides have the same base…

A

Solve the powers in a equation for x

31
Q

How to solve equations with e in it

A

Take the log base e of each side (not log 10)

32
Q

Graph of y=a^x as x changes

A

x<0, graph approaches x-axis faster

x>0 graph approaches y-axis faster (steeper)

33
Q

graph of y = ka^x

A

dilates the graph of y=a^x by a factor of k

34
Q

graph of y = ka^x + c

A

Shifts graph of y = ka^x up (+) or down (-)

35
Q

graph of y = ka^x+b

A

Shifts the graph of y = ka^x left (+) or right (-)

36
Q

graph of y=loga x as base a changes

A

as a increases:

01, graph approaches x-axis faster

37
Q

graph of y = klog a x

A

Dilates graph by factor of k (enlarged)

times all coordinated by factor k

38
Q

graph of y = kloga x + c

A

Shifts graph up (+) or down (-)

39
Q

graph of y = kloga (x+b)

A

Shifts graph left (+) or right (-)

40
Q

When drawing a graph from a log question…

A

x-axis - letter on the right

y-axis - letter on the left

41
Q

How to find equation of exponential given the graph

A
  1. Start with general equation y=a^x
  2. Look at asymptote to see if shifted up or down
  3. Sub a point given then solve to find a
42
Q

How to find equation of log given the graph

A
  1. Start with general equation y=loga x
  2. Look at asymptote to see if shifted up or down
  3. Sub point into equation to find a
43
Q

How to find the rate of change with logs/exponentials

A

find the derivative of the original equation then sub in value for t

Maths Prelim (1 decks)

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