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Math 30 > U2 A1- Exponents And Logs > Flashcards

U2 A1- Exponents And Logs Flashcards

1
Q

What are the restrictions on

y=ab^x

A

a can’t equal 0

b>0
b can’t equal 1

x is an element of the real numbers

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2
Q

Why can’t b=1 in y=ab^x

A

You get the same value regardless of what exponent it is (so you’d get a linear function)

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3
Q

Graph of decay functions: ___ from left to right

A

Fall

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4
Q

Graph of growth functions: ____ from left to right

A

Rise

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5
Q

What is the domain and range of an exponential function

A

Domain: (-infinite, infinite)

Range: (0,infinite)

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6
Q

What is the x intercept of an exponential function

A

None

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7
Q

What is the y intercept of an exponential function

A

(0,1)

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8
Q

What is the horizontal asymptote in exponential equations

A

y=0 (x axis)

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9
Q

Why is y=(1/2)^x a reflection about the y axis compared to

y=2^x

A

You can see that the base is flipped and in order to flip the base you have to make the exponent negative

Since x is now negative there’s a reflection about the y axis

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10
Q

In y=ab^x what determines whether the function rises or falls

A

b value

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11
Q

Exponential functions: b>1 means the function will ____

A

Rise

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12
Q

Exponential functions: 0<b></b>

A

Fall

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13
Q

Exponential functions: what does “a” represent

A

y intercept

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14
Q

What is the equation that represents an exponential function

A

y=ab^x

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15
Q

Exponential functions: as the value of b increases, the curve has a _____ rise

A

Steeper

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16
Q

Exponential functions: as the value of b decreases, the curve has a steeper ____

A

Fall

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17
Q

Exponential functions: when determining transformations do you still have to remember (x-h)

A

Yes

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18
Q

What is anything to the exponent 0

A

1

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19
Q

Exponential functions: a negative exponent causes the base to ____

A

Flip

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20
Q

What is the negative exponents law

A

(a/b)^-m = (b/a)^m

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21
Q

What is the power of a power law

A

(a^m)^n= a^m(n)

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22
Q

What is the rational exponents law

A

(n exponent root x)^m =

x^(m/n)

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23
Q

How can you solve an exponential equation graphically

A

y1= one side of the equation

y2= other side of the equation

point of intersection is the solution

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24
Q

Steps for solving exponential equations by changing the base

A

1) write both sides of the equation in the same base
2) use exponent laws so that each side of the equation only has 1 power
3) cancel out the bases
4) solve for x

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25
Q

If you’re solving an exponential equation by changing the bases how would you write this as a single exponent (step 3)

2^2x/2^3x+2

A

Concept 2 fill in

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26
Q

What is half life

A

Time needed for a substance to reduce to half its amount

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27
Q

What is the inverse of an exponential function

A

Logarithmic function

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28
Q

What are the five characteristics of a logarithmic function

A

D: (0,infinite)

R:( -infinite, infinite)

x intercept (1,0)

y-intercept: none

VA: x=0

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29
Q

How are the characteristics of a logarithmic function related to those of an exponential function

A

They are the inverse of each other

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30
Q

What is the base in log form

A

The subscript number

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31
Q

Where is the exponent written in log form

A

Other side of equals sign

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32
Q

When converting from exponential to log for the ____ and ____ switch

A

Input

Output

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33
Q

What is the argument in logarithmic form

A

Value after the subscript

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34
Q

If you flip the base, you must change the ____ of its exponent

A

Sign

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35
Q

What are the restrictions on logarithmic functions

y=logb(x)

A

b>0

b can’t equal 1

x>0

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36
Q

When determine vertical stretch factor what do you always have to make sure of

A

That y is ISOLATED

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37
Q

A logarithmic function is a reflection about the line _____

A

y=x

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38
Q

How to solve a logarithm if given variables instead of numbers

A

1) make an equals sign and put an x after it
2) convert to exponential form
3) find a common base and solve

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39
Q

In log form if the subscript and argument are the same value what is always the answer

A

1

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40
Q

What is log(sub5) (1/125)

Without a calculator

A

1) make an equals sign and say it equals x
2) concert to log form

3) must be a negative exponent since we have to flip the base and 5^3 is 125 so 5^-3 is
(1/125)

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41
Q

If the argument is one what is always the exponent (answer)

A

0

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42
Q

What is the east way to evaluate a log

A

Put it into calc

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43
Q

Log laws: when we add 2 logs in the same base we can _____ the arguments

A

Multiply

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44
Q

Log laws: when we subtract 2 logs in the same base we ____ the arguments

A

Divide

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45
Q

Log laws: when we have the log of a power we can take the exponent and move it _____ to become the ______

A

Upfront

Coefficient

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46
Q

Can log laws be used on logs that don’t have the same base

A

No

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47
Q

What is log(sub b) b^n

Always equal

A

n

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48
Q

When simplifying logs what do you have to do if there’s an coefficient in front

A

Move it back to the exponent

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49
Q

You cannot use logarithm law as to simplify if there’s a _____

A

Coefficient

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50
Q 
If log(sub 2)x=a  determine an expression in terms of “a” for 
log(sub 2)16x^2
A

So this means “a” is the only variable that can be in the answer

1) separate into addition statement (make sure the x’s stay together)
2) move exponent out front
3) now you can solve one part of the expression and put the other part ion terms of a

Answer: 4 + 2(a)

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51
Q

When given a long expression where you have to simplify using log laws how should you do it

A

Do it all at once (so addition terms are in the numerator and subtraction terms are in the denominator)

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52
Q

If there’s a variable with a negative exponent in numerator what can you do

A

Put the variable in the denominator and switch the sign of the exponent

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53
Q 
If log(sub 2)5=a  determine an expression in terms of “a” for 
log(sub 2)10
A

1) separate the 10 into 5(2)
2) write as an addition statement (log laws)
3) a+1 is answer

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54
Q

What is the solution to 16^(1/2) using the rational exponent law

A

Root 16 =4

Since the denom exponent goes out to the front and the num exponent applies to only the number

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55
Q

How do you know to use the rational exponent law

A

If exponent is a fraction

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56
Q

How to find the point of intersection on a graph

A

Hit 2nd, trace, intersect

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57
Q

What are the two options to solve exponential equations with unlike bases

A
  • convert to log form

- take the log of both sides

58
Q

How to solve exponential equations by taking the log of both sides

A

1) take the log of both sides
2) bring any exponents to the front
3) foil out
4) get the x terms on one side and numerical terms on the other side
5) factor out an x
6) isolate for x and solve

59
Q

How to solve exponential equations by converting to log form

A

1) Convert to log form

2) isolate for x

60
Q

How do you know which of the two methods to use when solving an exponential equation

A
  • if variable only on one side convert to log form

- if variable on both sides take the log

61
Q

Solving exponential equations by taking the log of both sides: can you usually cancel out the x terms when you put them all on one side

(And why)

A

No, because the arguments of each are usually diff

62
Q

Growth/decay applications: what do the variables represent represent in y=ab^x

A

y: final amount
a: initial amount
b: Growth/decay factor
x: variable

63
Q

Growth/decay applications: how do you find the value of b

A

1)Convert percent given value in question to decimal

2) add that value to 1 if it’s growth
subtract that value from 1 if it’s decay

64
Q

Growth/decay applications: why does b always start as 1

A

Since this is 100%, which represents no growth or decay

65
Q

Growth/decay applications: steps for solving

A

1) state values a and b
2) state the exponential function
3) sub in values
4) solve for x (using one of the known methods)

66
Q

If question states “each year” or “annually” what equation do you use

A

y=ab^x

67
Q

What is the b value in each of these: tripling time, doubling time, half life

A

3,2, (1/2)

68
Q

What do the variables represent in A(t)=A(sub o)b^(t/h)

A

A(t)= amount after time

A(sub o)= initial amount

b= growth/decay factor

t= time elapsed

h= time related to how fast the model is growing or decaying

69
Q

What is the difference between “reduced to” and “reduced by”

A

Reduced to means you use that given number

Reduced by means you use the subtracted number

70
Q

What is the compound interest formula

A

A=P(1+i)^n

71
Q

What do the variables represent in the formula A=P(1+i)^n

A

A= final amount

P=initial amount

i= interest rate per compounding period

n= # of compounding periods

72
Q

How do you write the cube root of 16 as an exponent

A

16^(1/3)

farthest left is the BOTTOM # in the exponent

73
Q

In the power of a power rule what do you have to remember (if the outside exponent is a binomial)

A

To distribute the inside exponent into ALL terms of the outside exponent

74
Q

How many weeks are in a year

A

52

75
Q

How to write the inverse of a log function (or exponential)

A

1) convert to other form

2) switch x and y

76
Q

How are exponential form and log form of the same function related to each other

A

They are reflections of each other about the line y=x

77
Q

How are the graphs of exponential form and log form of the same function similar

A

They have the same shape

78
Q

How are the graphs of exponential form and log form of the same function different

A

One is increasing and the other is decreasing

79
Q

Expanding using log laws: if an argument is a root what should you do

A

Change it to an exponent and bring the exponent out front

80
Q

Expanding using log laws: if there’s a common coefficient in you final answer what can you do

A

Factor it out

81
Q 
How would you expand 
logx root(y/z)
A

1) expand into addition

2) expand that other part into subtract but that whole term has to be in brackets

82
Q

Expanding/evaluating using log laws: WHAT DO YOU ALWYAS SO WITH THE COEFFICIENTS

A

Being them back as exponents

83
Q

What do you do if a whole log term is being divided by a term

A

Same as multiplying the whole term by the reciprocal so turn it into a fraction

84
Q

If a division sign is going through the logs (instead of just the arguments) can the term be seperated

A

No

85
Q

What translation do you need to get from y=log(sub 2)x to

y=log(sub 2)8x

A

3

Ask yourself: what exponent with a base of 2 will equal 8

86
Q

When expanding using log laws should you get rid of a root sign or keep it

A

Get rid of it (change to exponent)

87
Q

When stating restrictions do you always use < or line under and

A

ALWAYS USE

88
Q

How to solve: what is the diff between an 8.9 and 7.1 earthquake

A

Find the difference and make that an exponent with a base of 10 and solve

89
Q

y=a^x and y=(1/a)^x are reflections of each other in what axis

A

y axis

Since x becomes negative

90
Q

What is a logarithmic equation

A

Has at least one variable in the argument

91
Q

What is the restriction on the argument in a logarithmic equation

A

Argument>0

92
Q

What does option 1 look like for solving logarithmic equations

A

There are term(s) without logs

93
Q

Steps for solving logarithmic equations: option 1

A

1) Make sure all the log terms are on one side
2) use log laws to simplify that side
3) convert to exponential form
4) solve for x (should be either a rational function or quadratic)

94
Q

What does option 2 look like for solving logarithmic equations

A

All the terms have a log in them

95
Q

Steps for solving logarithmic equations: option 2

A

1) make sure all the terms with variables are on the same side
2) use log laws to simplify each side to only 1 logarithm
3) cancel the log on each side
4) solve for x (should be either rational function or quadratic)

96
Q

Steps for stating restriction on logarithmic equation

A

1) determine restrictions from original equation

2) x has to satisfy both so only pick the restriction where x would satisfy both of them

97
Q

How to determine solution(s) and extraneous roots(s) once you solve a logarithmic function

A

1) determine the restriction

2) see which of the possible solutions satisfies it

98
Q

If question mentions “doubles, triples, or half life” what formula do we use

A

A(t)= A(sub o) b^(t/h)

99
Q

Steps for solving equations with doubling, tripling, or half life

A

1) identify knowns and unknowns (remember b is either 3,2, or 1/2)
2) plug values into equation
3) divide both sides by initial amount (Ao)
4) take the log of both sides (since it’s an exponential equation
5) bring exponent out front
6) get rid of denominator
7) isolate for x

100
Q

What formula do you use to solve for growth/decay

A

y=ab^x

101
Q

How do you find the value of “n” in the compound interest formula

A

n= # of compounding periods per year(years)

102
Q

Compound interest formula: how do you know how many compounding periods there are (to find value of n)

A

Annually= 1 compounding period

monthly= 12 compounding periods

103
Q

How many compounding periods are there semi annually and quarterly

A

Semi annually=2

Quarterly=4

104
Q

How do you find the value of “i” in the compound interest formula

(And what do you have to remember to do with the interest rate)

A

i = interest rate/ # of compounding periods per year

HAVE TO CONVERT PERCENT INTEREST RATE TO A DECIMAL

105
Q

What variables require extra thinking to find in the compound interest formula

A

n and i

106
Q

In what type of questions do you add the percent value as a decimal to 1 to get the b value

A

Growth/ decay questions

107
Q

Compound interest formula: if “i” is the unknown what do you still HAVE to write it as

A

i = i/ # of compounding periods per year

108
Q

Steps for solving equations with the compound interest formula

A

1) identify knowns and unknowns (remember values n and i can’t be directly found in question)
2) plug values into equation
3) divide both sides by initial amount (P)
4) take the log of both sides (since it’s an exponential equation)
5) bring the exponent out front
6) isolate for x

109
Q

What does log(sub)b always equal

A

1

110
Q

What does log(sub b) b^n

always equal

A

n

111
Q

What does b^log(sub b)n

always equal

A

n

112
Q

When changing the base with a radical what should you always do first

A

Change it back into an exponent (to get rid of the root sign)

113
Q

If you’re having troubles with variables what can you always do

A

Plug in numbers

114
Q

WHAT SOULD YOY ALWAYS DO ONCE YOU HAVE AN ANSWER (that you found using the half life formula or compound interest formula)

A

Sub solution back into equation to see if you’re right

115
Q 
How can tot write 
m log(sub p)n=q in exponential form
A

1) bring m back as an exponent
2) convert to exponential form
3) isolate for n by take the root m of both sides

116
Q

If question is asking you to do stuff with variables how should you check your answer after

A

Find numbers that work and sub the numbers in to se did they work

117
Q

If the divisor sign is over the logs too can you seperate using log laws

A

No

118
Q

If question asks for the inverse of a function how do you always check your answer

A

Graph them to see if they’re a reflection of each other over the line y=x

119
Q

STEPS FOR FINDING THE INVERSE OF AN EQUATION

A

1) convert to exponential form
2) switch x and y
3) isolate for y (if needed)
4) check answer by graphing

120
Q

If trying to solve a logarithmic equation and the base is also unknown what do you do differently in your solving steps

A

Nothing, solve as you normally would

121
Q

If question asks for an EQUIVALENT equation how can you always check your answer

A

GRAPH BOTH FUNCTIONS AND THEY SHOULD BE THE SAME GRAPH

Sub in numbers to see if it works

122
Q

How to solve: if 3^logx =9

What is the value of x

A

1) 3^3=9, so 3^-3= 1/9
2) meaning logx= -2
3) convert above equation into exponential form and solve for x

123
Q

If b(c)=a what is the value of

Log(sub a)b + log(sub a)c

A

1) combine using log laws so you get b(c)=a

2) use that fact and the fact that the base is also “a” to determine the answer is 1

124
Q

The expression

y=log(sub a) (a^4 •b) - log(sub a) (ab)

A

1) combine using log laws into division statement

2) use exponent laws to subtract exponents of like bases and you end with 3

125
Q

If word problem is asking for years what do you have to remember to do with final answer

A

Round it UP (regardless of what the number is)

126
Q

If question says the ph of something is 4, what is the ph of something five times more acidic

Which substance is the greater number

A

Substance with ph 4 (since acids are lower on ph scale than bases)

127
Q

What do you have to remember with ph questions

A

Acid is 1-7 and base is 7-14

128
Q

For growth/decay questions what formula do you use

A

y=ab^x

129
Q

Compound interest questions: if question asks for the coefficient of “n” (in years) what do you have to remember to do IF the investment ISNT compounded yearly

A

Convert answer to years

Ex) investment is compounded monthly so divide final answer by 12 to get years

130
Q

How do you ALWAYS find the y intercept

A

Let x=0 and solve for y

131
Q

When finding domain and range how can you always check your answer

A

Graph with a calc

132
Q

If question asks for an equivalent equation and you’re getting stuck what can you try

(especially if options have log in the numerator and denominator)

A

Converting it into the log change base formula

133
Q

In the log change base formula what value is the numerator and what value is the denominator

A

Numerator- log argument

Denominator- log base

134
Q

Half life/ doubling time questions: if question gives you an exact value for half life what variable does this represent

(And why)

A

“h”

Since the half life is related to how fast the model is decaying

135
Q

If question gives you an exact value for half life is this the value of “b” and why

A

No, b is ALWAYS (1/2) for half life,

It would be the value of h

136
Q

If application question DOESN’T mention half life/doubling time/tripling time (and it’s not a compound interest formula) what formula do you use

(And what do you have to remember for the b value)

A

y=ab^x

add/ subtract value from 1 depending on if it’s growth or decay

137
Q

If question asks for range (and there’s a restriction) WHERE does the restriction alway have to be in the answer

(And why)

A

AFTER the comma (since x is always a restriction in the range)

(x,y)
(infinite, restriction)

138
Q

HOW DO YOU ALWAYS CHECK YOUR ANSWER TO AN EXPONENTIAL EQUATION

A

Graph y1 and y2 to see where they intersect

139
Q

Logarithmic scales (applications): do we graph the actual value or the exponent

A

Actual value

140
Q

Logarithmic scales (applications):

What is the formula for comparing two earthquakes

(And what do the variables represent)

A

I1/ I2 = 10^m1-m2

I= intensity

m=magnitude

141
Q

Logarithmic scales (applications):

What is the formula for comparing two sounds

(And what do the variables represent)

A

I1/I2 = 10^(dB1-dB2)/10

I=sound intensity

dB=decibal level

142
Q

Logarithmic scales (applications):

What is the formula for comparing ph

A

10^pH1/pH2

Math 30 (6 decks)

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