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32 Cards in this Set
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if y=log(x) then what =x?

10^y = 10^(log(x)) = x


log(100)=?

2


True or false, the expression log(10000000.....) always equals the number of zeros it contains

True.
examples: log(100)= 2 has two zeros log(1)=0 has zero zeros log(1000000)=6 has 6 zeros 

for any N, log(10^N)=?
i.e. log of 10 raised to the Nth power=? 
N.


What is wrong with this statement?
For any N, 10^(logN) = N 
N must be > 0.
If N>0 this statement is true 

for a and b >0 and any t,
what is: log(ab)? 
log(a) + log(b)


For a,b>0 and any t what is:
log(a/b)? 
log(a)log(b)


for a,b>0 and any t, what is
log(b^t)? 
t*log(b)


What are the steps for solving an equation where the variable is in the exponent and only on one side of the equation? For example, solve:
100*2^t=337,000,000 for t 
1. get the expression with the variable by itself on one side:
2^t=(337,000,000)/(100) 2. use log or ln to bring the variable down: log(2^t)=log(3,370,000) t*log(2)=log(3,370,000) 3. Get the variable by itself on one side and everything else on the other: t= (log(3,370,000)/(log(2)) = 21.684 

Does it matter if you use ln or log for a problem?

No, just make sure you apply the same function to both sides of the equation. That is, if you are taking the log of one side of an equation, you must take the log, and not the ln, of the other side of the equation.


ln(e^x)=?

x


for all x>0,
e^(ln(x))=? 
x


simplify the expressions:
1. ln(a) + ln(b) 2. ln(a) ln(b) 3. t*ln(b) 
1. ln(ab)
2. ln(a/b) 3. ln(b^t) 

True or false,
ln(x) and log(x) basically have the same algebraic properties: for example, log(ab)= log(a) +log(b) and ln(ab)= ln(a) + ln(b) 
True


by definition, y=ln(x) means

e^y = x


What steps would you take to solve a problem involving the number e being raised to the variable for which you want to solve?
For example: Solve for x 5e^(2x) = 50 
1. get e^(2x) by itself on one side by dividing each side by 5:
e^(2x)=10 2. take the ln of both sides to bring down the exponent: ln(e^(2x))= ln(10) 2x=ln(10) 3. Get the variable, x, by itself on one side and everything else on the other: x= ln(10)/2= 1.151 

To solve problems where the variable is in the exponent you want to use......

logs or ln to bring the variable down from the exponent.


What 5 properties do the graphs of ln(x) and log(x) have in common?

The graphs are very similar:
1. domain of both is x>0 2. range of both is all reals 3. both have vertical asymptote at x=0 4. both are slowly increasing 5. both are concave down 

What is the inverse of y=log(x)?

the inverse of y=log(x) is y=10^x


What is the inverse of y=ln(x)

the inverse of y=ln(x) is y=e^x


As the x values get closer and closer to zero and are approaching from the right (i.e. going from 0.1 to 0.01 to 0.001... rather than 0.1 to 0.01 to0.001 etc) then the graph of both ln(x) and log(x) do what?

shoot down toward negative infinity, staying close the the x=0 line


true or false, both ln(x) and log(x) have horizontal asymptotes?

False, both graphs may appear to have horizontal asymptotes but in reality they are both increasing very slowly and are not bounded by any line y=k.


The graph of f has a horizontal asymptote at y=a if.......

f(x)> a as x> infinity or negative infinity
(as f(x) approaches a, x approaches positive or negative infinity) 

The graph of f has a vertical asymptote at x=a if......

f(x)>infinity or negative infinity as x> a


What are the asymptotes of the function y=log(x2)?

Since y=log(x2) is just a horizontal shift by 2 to the right of log(x), then the vertical asymptote of log(x2) is the vertical asymptote of log(x) shifted two to the right. Since the vertical asymptote of log(x) is x=0, the vertical asymptote of y=log(x2) is x=2. there are no horizontal asymptotes.


What are the asymptotes of the function e^(x)?

We know y=e^x has a horizontal asymptote at y=0.
Since y=e^(x) is the graph of y=e^x when it has been flipped across the y and then the x axis, we should flip the horizontal asymptote over the y then x axis: Thus flip y=0 over y axis gives us still y=0. Flipping over x axis gives us still y=0. Thus the horizontal asymptote is still at y=0. Can graph this to convince yourself. 

true or false, if a and b are >0 then ln(a + b)= ln(a) +ln(b)

false, ln(a+b) cannot be simplified


true or false, log(a/b) = log(a)/log(b)

false, log(a/b)= log(a) log(b)


The values of log(10) and ln(e) =?

1


Correct this statement:
If 7.32 = e^t, then t=7.32/e 
you must use ln to bring the t down.
7.32 = e^t ln(7.32)= ln(e^t) ln(7.32)=t 

If a population doubles in size after 20 years, what is its continuous growth rate?

y=ae^(kt)
2a=ae^(k*20) 2=e^(k*20) ln(2)= ln(e^(20k)) ln(2)= 20k k=ln(2)/20= 0.035 is the cont. growth rate 

If the halflife of a substance is 5 hrs then there will be what % of the substance left after 25 hrs?

each 5 hrs 1/2 of the substance is removed:
let c be the initial amount of substance. then after 5hrs we have 0.5c left after 10 hrs we have half of 0.5c= 0.25c. After 15 hrs we have half of 0.25c which is 0.125c after 20 hrs we have half or 0.125c which is 0.0625c after 25 hrs we have half of 0.0625c which is 0.03125c. So we have about 3.125% of c left 