Chapter 4- Logarithmic Functions

Front 1

if y=log(x) then what =x?

Back 1

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

Front 2

log(100)=?

Back 2

2

Front 3

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

Back 3

True.

examples:

log(100)= 2 has two zeros
log(1)=0 has zero zeros
log(1000000)=6 has 6 zeros

Front 4

for any N, log(10^N)=?

i.e. log of 10 raised to the Nth power=?

Back 4

N.

Front 5

What is wrong with this statement?

For any N,
10^(logN) = N

Back 5

N must be > 0.
If N>0 this statement is true

Front 6

for a and b >0 and any t,
what is:

log(ab)?

Back 6

log(a) + log(b)

Front 7

For a,b>0 and any t what is:

log(a/b)?

Back 7

log(a)-log(b)

Front 8

for a,b>0 and any t, what is

log(b^t)?

Back 8

t*log(b)

Front 9

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

Back 9

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

Front 10

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

Back 10

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.

Front 11

ln(e^x)=?

Back 11

x

Front 12

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

Back 12

x

Front 13

simplify the expressions:

1. ln(a) + ln(b)

2. ln(a)- ln(b)

3. t*ln(b)

Back 13

1. ln(ab)
2. ln(a/b)
3. ln(b^t)

Front 14

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)

Back 14

True

Front 15

by definition, y=ln(x) means

Back 15

e^y = x

Front 16

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

Back 16

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

Front 17

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

Back 17

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

Front 18

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

Back 18

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

Front 19

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

Back 19

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

Front 20

What is the inverse of y=ln(x)

Back 20

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

Front 21

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 to-0.001 etc) then the graph of both ln(x) and log(x) do what?

Back 21

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

Front 22

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

Back 22

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.

Front 23

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

Back 23

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

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

Front 24

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

Back 24

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

Front 25

What are the asymptotes of the function y=log(x-2)?

Back 25

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

Front 26

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

Back 26

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.

Front 27

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

Back 27

false, ln(a+b) cannot be simplified

Front 28

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

Back 28

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

Front 29

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

Back 29

1

Front 30

Correct this statement:

If 7.32 = e^t, then t=7.32/e

Back 30

you must use ln to bring the t down.

7.32 = e^t
ln(7.32)= ln(e^t)
ln(7.32)=t

Front 31

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

Back 31

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

Front 32

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

Back 32

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