Chapter 3 (3.1 to 3.5)

Question 1

What is the one to one property and how could it be used to solve 16 = 2^x+2?

Answer 1

When two numbers are equal to each other and their bases are equal, their exponents must also be equal (2^4 = 2^x+2 so 4 = x + 2 so x is 2)

Question 2

What is the “natural exponential function”?

Answer 2

Y = e^x

Question 3

What is the formula for exponential functions and how can it be moved right, left, up, down, or flipped over either axis?

Answer 3

Y = a^x
Right is (x - a number), Left is (x + a number), Up is (a^x) plus a number, Down is (a^x) minus a number, to flip over the x multiply a by -1, and to flip over y multiply x by -1

Question 4

What is the formula for compounding interest?

Answer 4

P (1 + (r/n))^r*t
P is principle
R is rate in a decimal
N is number of times it is compounded in a year
T is number of years

Question 5

What is the formula for continuously compounding interest?

Answer 5

Pe^(rt)
P is principle
R is rate in a decimal
T is number of years

Question 6

What is the definition of a log?

Answer 6

An exponent which raises a base to get a given number

Question 7

What is log form, how can it be changed to exponent form, and what are the limitations?

Answer 7

Log(b) X = Y if b^y = X

Question 8

How would you change Log(2) 32 = 5 to exponent form?
How would you change 10^3 = 1000 to log form?
How would you change e^2 = 7.389 to log form?

Answer 8

2^5 = 32
Log 1000 = 3 (log with no subscript is base 10)
Ln 7.389 = 2 (Ln is log(e))

Question 9

What is log(b) 1?

Answer 9

0 (b to the zero is one)

Question 10

What is log(b) b?

Answer 10

1 (b to the first is one)

Question 11

What is log(b) b^x?

Answer 11

X (b to the x is b to the x)

Question 12

What is b^(log(b) x)

Answer 12

X (if you set change it to log form you get log(b) Y = Log(b) X so y must equal x and therefore the problem, that was set to equal y also equals x)

Question 13

What do you know if log(b) x = log(b) y

Answer 13

X=Y (this is called the one to one property and is used a lot to find an x)

Question 14

What does log(b) MN equal and what is the proof?

Answer 14

Log(b) M + Log(b) N
set log(b) MN to z, log(b) M to x and log(b) N to y
then set them to exponent form (b^z = MN, b^x = M and b^y = N) then substitute (b^x = b^y times b^x) then use the one to one property of exponents (x = y + x) then substitute

Question 15

What does log(b) M/N equal and what is the proof?

Answer 15

Log(b) M - Log(b) N
set log(b) MN to z, log(b) M to x and log(b) N to y
then set them to exponent form (b^z = M/N, b^x = M and b^y = N) then substitute (b^x = b^y / b^x) then use the one to one property of exponents (x = y - x) then substitute

Question 16

What does log(b) M^k equal and what is the proof?

Answer 16

K * Log(b) M
Log(b) M^k is log(b) MMM*M k times, and you can change that to lob(b) M + lob(b) M + lob(b) M k times then change that to K times Log(b) M

Question 17

What is the change of base formula and what is the proof?

Answer 17

Log(b) X is log X/lob B (or both Ln)
Set Log(b) X equal to y, then change it to exponent form (b^y = x), then log/ln both sides (Log b^y = Log x), then simplify the first one (Y * Log(b)) and divide off to get y which is log X/lob B

Question 18

How would you solve e^x = 13?

Answer 18

Ln 13 is x

Question 19

How would you solve Ln x = -8?

Answer 19

e^-8 is x

Question 20

How would you solve Log x = -2?

Answer 20

10^-2 is x

Question 21

How would you solve 3^x = 20?

Answer 21

Log(3) 20 is x

log 20/Log 3 is x