U2 A1- Exponents And Logs Flashcards
What are the restrictions on
y=ab^x
a can’t equal 0
b>0
b can’t equal 1
x is an element of the real numbers
Why can’t b=1 in y=ab^x
You get the same value regardless of what exponent it is (so you’d get a linear function)
Graph of decay functions: ___ from left to right
Fall
Graph of growth functions: ____ from left to right
Rise
What is the domain and range of an exponential function
Domain: (-infinite, infinite)
Range: (0,infinite)
What is the x intercept of an exponential function
None
What is the y intercept of an exponential function
(0,1)
What is the horizontal asymptote in exponential equations
y=0 (x axis)
Why is y=(1/2)^x a reflection about the y axis compared to
y=2^x
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
In y=ab^x what determines whether the function rises or falls
b value
Exponential functions: b>1 means the function will ____
Rise
Exponential functions: 0<b></b>
Fall
Exponential functions: what does “a” represent
y intercept
What is the equation that represents an exponential function
y=ab^x
Exponential functions: as the value of b increases, the curve has a _____ rise
Steeper
Exponential functions: as the value of b decreases, the curve has a steeper ____
Fall
Exponential functions: when determining transformations do you still have to remember (x-h)
Yes
What is anything to the exponent 0
1
Exponential functions: a negative exponent causes the base to ____
Flip
What is the negative exponents law
(a/b)^-m = (b/a)^m
What is the power of a power law
(a^m)^n= a^m(n)
What is the rational exponents law
(n exponent root x)^m =
x^(m/n)
How can you solve an exponential equation graphically
y1= one side of the equation
y2= other side of the equation
point of intersection is the solution
Steps for solving exponential equations by changing the base
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
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
Concept 2 fill in
What is half life
Time needed for a substance to reduce to half its amount
What is the inverse of an exponential function
Logarithmic function
What are the five characteristics of a logarithmic function
D: (0,infinite)
R:( -infinite, infinite)
x intercept (1,0)
y-intercept: none
VA: x=0
How are the characteristics of a logarithmic function related to those of an exponential function
They are the inverse of each other
What is the base in log form
The subscript number
Where is the exponent written in log form
Other side of equals sign
When converting from exponential to log for the ____ and ____ switch
Input
Output
What is the argument in logarithmic form
Value after the subscript
If you flip the base, you must change the ____ of its exponent
Sign
What are the restrictions on logarithmic functions
y=logb(x)
b>0
b can’t equal 1
x>0
When determine vertical stretch factor what do you always have to make sure of
That y is ISOLATED
A logarithmic function is a reflection about the line _____
y=x
How to solve a logarithm if given variables instead of numbers
1) make an equals sign and put an x after it
2) convert to exponential form
3) find a common base and solve
In log form if the subscript and argument are the same value what is always the answer
1
What is log(sub5) (1/125)
Without a calculator
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)
If the argument is one what is always the exponent (answer)
0
What is the east way to evaluate a log
Put it into calc
Log laws: when we add 2 logs in the same base we can _____ the arguments
Multiply
Log laws: when we subtract 2 logs in the same base we ____ the arguments
Divide
Log laws: when we have the log of a power we can take the exponent and move it _____ to become the ______
Upfront
Coefficient
Can log laws be used on logs that don’t have the same base
No
What is log(sub b) b^n
Always equal
n
When simplifying logs what do you have to do if there’s an coefficient in front
Move it back to the exponent
You cannot use logarithm law as to simplify if there’s a _____
Coefficient
If log(sub 2)x=a determine an expression in terms of “a” for log(sub 2)16x^2
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)
When given a long expression where you have to simplify using log laws how should you do it
Do it all at once (so addition terms are in the numerator and subtraction terms are in the denominator)
If there’s a variable with a negative exponent in numerator what can you do
Put the variable in the denominator and switch the sign of the exponent
If log(sub 2)5=a determine an expression in terms of “a” for log(sub 2)10
1) separate the 10 into 5(2)
2) write as an addition statement (log laws)
3) a+1 is answer
What is the solution to 16^(1/2) using the rational exponent law
Root 16 =4
Since the denom exponent goes out to the front and the num exponent applies to only the number
How do you know to use the rational exponent law
If exponent is a fraction
How to find the point of intersection on a graph
Hit 2nd, trace, intersect
What are the two options to solve exponential equations with unlike bases
- convert to log form
- take the log of both sides
How to solve exponential equations by taking the log of both sides
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
How to solve exponential equations by converting to log form
1) Convert to log form
2) isolate for x
How do you know which of the two methods to use when solving an exponential equation
- if variable only on one side convert to log form
- if variable on both sides take the log
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)
No, because the arguments of each are usually diff
Growth/decay applications: what do the variables represent represent in y=ab^x
y: final amount
a: initial amount
b: Growth/decay factor
x: variable
Growth/decay applications: how do you find the value of b
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
Growth/decay applications: why does b always start as 1
Since this is 100%, which represents no growth or decay
Growth/decay applications: steps for solving
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)
If question states “each year” or “annually” what equation do you use
y=ab^x
What is the b value in each of these: tripling time, doubling time, half life
3,2, (1/2)
What do the variables represent in A(t)=A(sub o)b^(t/h)
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
What is the difference between “reduced to” and “reduced by”
Reduced to means you use that given number
Reduced by means you use the subtracted number
What is the compound interest formula
A=P(1+i)^n
What do the variables represent in the formula A=P(1+i)^n
A= final amount
P=initial amount
i= interest rate per compounding period
n= # of compounding periods
How do you write the cube root of 16 as an exponent
16^(1/3)
farthest left is the BOTTOM # in the exponent
In the power of a power rule what do you have to remember (if the outside exponent is a binomial)
To distribute the inside exponent into ALL terms of the outside exponent
How many weeks are in a year
52
How to write the inverse of a log function (or exponential)
1) convert to other form
2) switch x and y
How are exponential form and log form of the same function related to each other
They are reflections of each other about the line y=x
How are the graphs of exponential form and log form of the same function similar
They have the same shape
How are the graphs of exponential form and log form of the same function different
One is increasing and the other is decreasing
Expanding using log laws: if an argument is a root what should you do
Change it to an exponent and bring the exponent out front
Expanding using log laws: if there’s a common coefficient in you final answer what can you do
Factor it out
How would you expand logx root(y/z)
1) expand into addition
2) expand that other part into subtract but that whole term has to be in brackets
Expanding/evaluating using log laws: WHAT DO YOU ALWYAS SO WITH THE COEFFICIENTS
Being them back as exponents
What do you do if a whole log term is being divided by a term
Same as multiplying the whole term by the reciprocal so turn it into a fraction
If a division sign is going through the logs (instead of just the arguments) can the term be seperated
No
What translation do you need to get from y=log(sub 2)x to
y=log(sub 2)8x
3
Ask yourself: what exponent with a base of 2 will equal 8
When expanding using log laws should you get rid of a root sign or keep it
Get rid of it (change to exponent)
When stating restrictions do you always use < or line under and
ALWAYS USE
How to solve: what is the diff between an 8.9 and 7.1 earthquake
Find the difference and make that an exponent with a base of 10 and solve
y=a^x and y=(1/a)^x are reflections of each other in what axis
y axis
Since x becomes negative
What is a logarithmic equation
Has at least one variable in the argument
What is the restriction on the argument in a logarithmic equation
Argument>0
What does option 1 look like for solving logarithmic equations
There are term(s) without logs
Steps for solving logarithmic equations: option 1
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)
What does option 2 look like for solving logarithmic equations
All the terms have a log in them
Steps for solving logarithmic equations: option 2
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)
Steps for stating restriction on logarithmic equation
1) determine restrictions from original equation
2) x has to satisfy both so only pick the restriction where x would satisfy both of them
How to determine solution(s) and extraneous roots(s) once you solve a logarithmic function
1) determine the restriction
2) see which of the possible solutions satisfies it
If question mentions “doubles, triples, or half life” what formula do we use
A(t)= A(sub o) b^(t/h)
Steps for solving equations with doubling, tripling, or half life
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
What formula do you use to solve for growth/decay
y=ab^x
How do you find the value of “n” in the compound interest formula
n= # of compounding periods per year(years)
Compound interest formula: how do you know how many compounding periods there are (to find value of n)
Annually= 1 compounding period
monthly= 12 compounding periods
How many compounding periods are there semi annually and quarterly
Semi annually=2
Quarterly=4
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)
i = interest rate/ # of compounding periods per year
HAVE TO CONVERT PERCENT INTEREST RATE TO A DECIMAL
What variables require extra thinking to find in the compound interest formula
n and i
In what type of questions do you add the percent value as a decimal to 1 to get the b value
Growth/ decay questions
Compound interest formula: if “i” is the unknown what do you still HAVE to write it as
i = i/ # of compounding periods per year
Steps for solving equations with the compound interest formula
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
What does log(sub)b always equal
1
What does log(sub b) b^n
always equal
n
What does b^log(sub b)n
always equal
n
When changing the base with a radical what should you always do first
Change it back into an exponent (to get rid of the root sign)
If you’re having troubles with variables what can you always do
Plug in numbers
WHAT SOULD YOY ALWAYS DO ONCE YOU HAVE AN ANSWER (that you found using the half life formula or compound interest formula)
Sub solution back into equation to see if you’re right
How can tot write m log(sub p)n=q in exponential form
1) bring m back as an exponent
2) convert to exponential form
3) isolate for n by take the root m of both sides
If question is asking you to do stuff with variables how should you check your answer after
Find numbers that work and sub the numbers in to se did they work
If the divisor sign is over the logs too can you seperate using log laws
No
If question asks for the inverse of a function how do you always check your answer
Graph them to see if they’re a reflection of each other over the line y=x
STEPS FOR FINDING THE INVERSE OF AN EQUATION
1) convert to exponential form
2) switch x and y
3) isolate for y (if needed)
4) check answer by graphing
If trying to solve a logarithmic equation and the base is also unknown what do you do differently in your solving steps
Nothing, solve as you normally would
If question asks for an EQUIVALENT equation how can you always check your answer
GRAPH BOTH FUNCTIONS AND THEY SHOULD BE THE SAME GRAPH
Sub in numbers to see if it works
How to solve: if 3^logx =9
What is the value of x
1) 3^3=9, so 3^-3= 1/9
2) meaning logx= -2
3) convert above equation into exponential form and solve for x
If b(c)=a what is the value of
Log(sub a)b + log(sub a)c
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
The expression
y=log(sub a) (a^4 •b) - log(sub a) (ab)
1) combine using log laws into division statement
2) use exponent laws to subtract exponents of like bases and you end with 3
If word problem is asking for years what do you have to remember to do with final answer
Round it UP (regardless of what the number is)
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
Substance with ph 4 (since acids are lower on ph scale than bases)
What do you have to remember with ph questions
Acid is 1-7 and base is 7-14
For growth/decay questions what formula do you use
y=ab^x
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
Convert answer to years
Ex) investment is compounded monthly so divide final answer by 12 to get years
How do you ALWAYS find the y intercept
Let x=0 and solve for y
When finding domain and range how can you always check your answer
Graph with a calc
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)
Converting it into the log change base formula
In the log change base formula what value is the numerator and what value is the denominator
Numerator- log argument
Denominator- log base
Half life/ doubling time questions: if question gives you an exact value for half life what variable does this represent
(And why)
“h”
Since the half life is related to how fast the model is decaying
If question gives you an exact value for half life is this the value of “b” and why
No, b is ALWAYS (1/2) for half life,
It would be the value of h
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)
y=ab^x
add/ subtract value from 1 depending on if it’s growth or decay
If question asks for range (and there’s a restriction) WHERE does the restriction alway have to be in the answer
(And why)
AFTER the comma (since x is always a restriction in the range)
(x,y)
(infinite, restriction)
HOW DO YOU ALWAYS CHECK YOUR ANSWER TO AN EXPONENTIAL EQUATION
Graph y1 and y2 to see where they intersect
Logarithmic scales (applications): do we graph the actual value or the exponent
Actual value
Logarithmic scales (applications):
What is the formula for comparing two earthquakes
(And what do the variables represent)
I1/ I2 = 10^m1-m2
I= intensity
m=magnitude
Logarithmic scales (applications):
What is the formula for comparing two sounds
(And what do the variables represent)
I1/I2 = 10^(dB1-dB2)/10
I=sound intensity
dB=decibal level
Logarithmic scales (applications):
What is the formula for comparing ph
10^pH1/pH2
