### Formula for interest compounded n times per year

A=P(1+r/n)^(nt)

A= balance

P= prinicpal

t= time in years

n= number of times compounded

r= interest rate (in decimal form)

### Formula for interest compounded continuously

A=Pe^(rt)

A= balance

P= prinicpal

t= time in years

r= interest rate (in decimal form)

### Describe the transformation f(x)=cb^x

Vertical Stretch c>1

Vertial Shrink 0<c<1

Multiply the y-coordinate of each point by "c"

### Logarithm

the exponent required to produce a given number, this of a positive number y to the base b is defined as follows: If y=b^x, then log b y=x

### Population

Find the percent increase or decrease, add or subtract from one, find a when x=0 and write equation - or set up two equations like y=ab^x

### Half Life

Find constant (.5N = N0e^k x half-life) Substitute real left over percentage for the .5 to find the years.

### Waves

Find scientific notation for large number and take log of number in order to place on number line

### Finding time difference

take number and raise 10 to that number, divide/subtract it from the other number 10 is raised to - use new number to determine how many times

### Hydrogen Ion Concentration

take pH and make it the negative exponent of 10, enter into calculator to find scientific notation

### Times difference between pH's

take -log of all scientific notations raise 10 to the power of pH's and subtract lower from higher. Raise 10 to the resulting number to find times.

### Exponential Function

a function of the form f(x)=ab×, where the coefficient a≠0, the base b>0 and b≠1

### Inverse Function: First Definition

two function f(x) and g(x) are inverse functions of each other if both are one-to-one functions and for every element in their domain f[g(x)]=g[f(x)]=x. The symbol for the inverse function is f(x) is f^-1(x)

### Inverse Function: Second Definition

for any one-to-one function f(x), its inverse, f^-1(x), is defined by the following statement: (a,b)is contained in f(x) if and only if (b,a) is contained in f^-1(x)

### Inverse Function: Third Definition

The one-to-one functions f(x) and g(x) are each other's inverses if and only if their graphs are symmetric with respect to the diagonal line f(x)=x

### exponential function

Includes a constant raised to a variable power, f(x) = b^x. The base b must be positive but cannot equal 1

### equivalent equations

All values for x and y that make one equation true also make the other one true ( b^x = b^y if and only if x=y)

### product rule for logarithms

states that the logarithm of a product of numbers equals the sum of the logarithms of the factors (log2 4*8 = ?)

### quotient rule for logarithms

states that the logarithm of the quotient of two numbers equals the difference of the logarithms of those numbers (log3 81/3 = ?)

### power rule of logarithms

states that the logarithm of a power of M can be calculated as the product of the exponent and the logarithm of M (log2 8^16 = ?)

### sound level

measured in units called decibels (dB); provides a scale that relates how humans perceive sound to a physical measure of its power

### Napierian logarithm

AKA natural logarithm, named after John Napier, a Scottish theologian and mathematician who discovered logarithms

### Newton's law of cooling

According to this law, the falling temperature obeys an exponential equation (y = ae^cx + T0, where T0 is the temperature surrounding the cooling object , x is the amount of time, and y is the current temperature)

### percent change in words

how much a quantity increases or decreases with respect to the original amount

### compound interest

interest is payed on the principal and previous interest payments

usually compounded at regular intervals

### Rate of Change

the percent increase or decrease of an exponential function, (the distance b is away from zero)

### Horizontal Asymptote

Imaginary line that acts as the lowest boundary of an exponential function, since it gets very close to zero, but not exactly zero

### solving exponential equations w/ variables

-move the exponent in front of the log by the power rule

-take the log of both sides to get x by istelf

### increases toward infinity

In a standard exponential function y = 2^x, when x increases toward infinity, y ___.

### increases toward zero

In a standard exponential function y = 2^x, when x decreases towards negative infinity, y ___.

### Logarithm - Definition

The inverse of taking the exponent of something

log₃27 =3

3³ =27

log₈1/64 = -2

8⁻² = 1/64

### Logarithm Property 5

= log₂√32/√8

= log₂(32/√8)¹/²

=1/2 ∗ log₂(32/√8)

= 1/2(log₂32 -1/2 ∗ log₂8)

= log₂32 - 1/4 ∗ log₂8

= 5/2 - 3/4 = 7/4

###
if logxA=logxB, than a=b

EX:log₆x+log₆(x-9)=2

log₆x(x-9)=2

x(x-9)=6²

x²-9x-36=0

(x-12)(x+3)=0

x-12=0 or x+3=0

x=12 x=-3

Check which anwser works in the problem

in the calculator.

Equality Property

###
A=PErt

A=Amount after T years

P=Principal (original) amount intrested

R=Annual interest rate

EX.

A=3000e^(0.04)(10)

A=3000e^0.4

A≈$4475.47

DON'T FORGET TO PUT THE DECIMAL PLACES INTO THE HUNDREDTH PLACE FOR MONEY!!!!

Compound Interest Formula

### property of equality

you use this when solving an equation: you must do the same thing to both sides of an equation

### Formula for interest compounded n times per year

A=P(1+r/n)^(nt)

A= balance

P= prinicpal

t= time in years

n= number of times compounded

r= interest rate (in decimal form)

### Formula for interest compounded continuously

A=Pe^(rt)

A= balance

P= prinicpal

t= time in years

r= interest rate (in decimal form)

### Describe the transformation f(x)=cb^x

Vertical Stretch c>1

Vertial Shrink 0<c<1

Multiply the y-coordinate of each point by "c"

### what does n^5 mean?

That n is multiplied by itself 4 times (n x n x n x n x n); note this is not 5 because n^1 is multiplied x 1, n^2 is multiplied by itself once (n x n), etc.

### a^m x a^n =

a^ (m+n) // to multiply numbers with the same base, keep the base and ADD the indices -p. 386

### a^(-1) =

1/a // remember that an exponent changes its sign when you move from numerator to denominator and back again -387

### a^(1/n) =

nth root of a // see p. 389 for the notation; note that the square root of a number is the same as that number raised to the power of 1/2.

### exponential function definition

a function in which the variable occurs as part of the index or exponent; simplest have the form f(x) = a^x where a is positive and not equal to 1.

### When you put money in the bank, what do you call your balance? What do you call the money the bank pays you? What do you call the effect of this growth?

Your balance is your PRINCIPAL; what the bank pays you is INTEREST; how the money grows is COMPOUNDED since the interest is then itself added to the principal which then earns you more interest -395

### What is the formula for the future value of an amount initially invested (present value) compounded at an interest rate per year of i for n years?

Future Value = Present Value x (1 + i) ^ n [396]

### If you invest $5000 at 8% for 2 years, how much money will you have?

FV = PV x (1+i)^n

FV = 5000 x (1 + .08) ^ 2 = 5000 x (1.08)^2 = $5,832 -396

### If I must save $10,000 for a boat in 4 years, and can get 8.5% per year return on my initial investment, how much should I invest today?

FV = PV x (1+i) ^n

10,000 = PV x (1.085)^4

Solve for PV => PV = $7,215.74

### What is the formula for depreciation?

Future Value = Present Value x (1 - i) ^ n [399] This is the mirror image of the future value of an amount compounded, it's just that a depreciation rate is just a NEGATIVE interest rate.

### What is a logarithm?

A logarithm in base 10 of any positive number is the power you would have to raise 10 to get that number. So the log of 100 is 2 since 10^2 is 100. The log of 1,000 is 3 and the log of 1,000,000 is 6 (for this special case, you can just count the zeros). -404

### Formula for interest compounded n times per year

A=P(1+r/n)^(nt)

A= balance

P= prinicpal

t= time in years

n= number of times compounded

r= interest rate (in decimal form)

### Formula for interest compounded continuously

A=Pe^(rt)

A= balance

P= prinicpal

t= time in years

r= interest rate (in decimal form)

### Describe the transformation f(x)=cb^x

Vertical Stretch c>1

Vertial Shrink 0<c<1

Multiply the y-coordinate of each point by "c"