Algebra 2, Lap 5: Exponential and Logarithmic Functions

How to find points of the basic function:

(-1, 1/B), (0,1), (1, B) (so you would identify the basic function, find the base (B), and use this B to find points on the basic function that can be used to apply the coordinate rule to)

what is the model for exponential decay?

* (1-r) must be >0 and <1

What is the parent equation for an exponential function?

- F(x)=Bˣ - B must be positive - B cannot be 1 - 0 < B < 1 (for decay), or B>1(for growth)

What is a logarithm?

-A logarithm is a way to solve for an exponent, rather than for the result -Because logs and their corresponding exponential expressions are equivalent, the values for logs have to follow the same rules as the values for exponentials (b must be positive and can't equal 1), and the answer portion cannot be negative

How to graph exponential functions (either growth or decay):

1. Find the most basic version of the function and Identify the base (B) 2. Find the points of the basic function using on the base 3. Identify the parts of the wanted function (a, k, b, B, x, y, h) 4. Find the coordinate rule 5. Apply the coordinate rule to the points of the basic function to find the points of the wanted function 6. Graph the points 7. Graph the asymptote 8. Connect the points

How to graph natural base functions:

1. Identify the Base (e) 2. determine if it can be growth or decay (is a positive?) 3. If so, then determine which one (e^b >1 is growth, 0< e^b < 1 is decay) 4. The points on f(x)=e^x are (0,1) and (1,e) 5. Find the coordinate rule and apply it to these parent points to find the transformed points for the function 6. The asymptote is at y=k 7. draw graph

graphing logarithmic functions

1. Identify the Base of the log 2. Because logs and exponentials are inverses, the parent points will be (1,0) and (B, 1) 3. Use the coordinate rule to transform the points 4. A vertical asymptote will be at x=h 5. graph the points

How to solve log equations

1. If there's only one log, then convert to an exponential expression 2. If there's multiple logs then try the one to one property 3. If that doesn't work, then try expanding or condensing to get it to work 4. If this still doesn't work, then take the ln of both sides, use the power property, distribute, then solve

how to convert percentages to decimals:

1. Percent to decimal = divide by 100 2. Decimal to percent = multiply by 100

1. if f(x)=aˣ, a>1, then the graph is __________ 2. if f(x)=aˣ, 0<a<1, then the graph is __________

1. growth 2. decay [note: you have to raise a to any exponent (the b value) before determining if it is growth or decay because a⁻ˣ = (a⁻¹)ˣ = (1/a)ˣ ]

Special properties of logarithms:

1. product property: If you have a log of a product, it can be split into the sum of the logs of each of the factors 2. Quotient property: If you have a log of a quotient, it can be split into the difference of the logs of the numerator and denominator 3. Power property: if you have a log of a power, the exponent can become a coefficient (these three above work in both directions to either expand or condense logs) 4. If the base and the "of" portion of a log match, then the log equals the exponent on the "of" portion 5. If a number is raised to a log with a base that matches the number, then the whole thing equals the "of" portion of the log

What is an asymptote?

A line which the graph approaches but never reaches (noted by a dashed line on a graph)

What is the formula for continuously compounded interest?

A= the final amount P= principal r= annual interest rate as a decimal t= time (years)

All logarithmic functions have a range of

All real numbers

What is exponential growth?

Exponential growth is when a quantity increases by the same percent in each unit of time (an exponential function is growth when B^b>1)

How to do tax word problems:

Original price + (tax rate x original price) = total price *note: whenever using tax rate in an equation, ALWAYS convert to decimal form

change of base formula

Used to evaluate a logarithm with any base (converts from one base to another)

Coordinate rule

[(⅟𝔟)x + h, ay+k] (remember, the equation for exponential functions is y= (a)Bᵇ⁽ˣ⁻ʰ⁾+k)

What is an exponential function?

a function in which x is the exponent

All exponential functions have a domain of

all real numbers

x=log𝔟(y)

bˣ=y

While logs can have any base, what are the two special bases?

common log = log base 10 = log natural log = log base e = ln

Within exponential growth, what are the two types of models?

compound interest and simple interest

While any number can by the base B of an exponential function, what special base is used for a natural base function?

e

What number is e?

e is approximately equal to 2.718 and is defined as: as n→∞, (1+(1/n))^n → e

What is exponential decay?

exponential decay is when a quantity decreased by the same percent in each unit of time (an exponential function is decay when 0 < B^b < 1)

What are the two types of exponential functions?

exponential growth and exponential decay (note: an exponential function is only a growth or decay if the a value is positive!!!)

Equation for Logarithmic functions:

f(x) = (a) logB(b(x-h)) + k

One-to-one property

for exponentials: if they have the same base, then the exponents are equal for logs: if they have the same base, then the "of" portions are equal

what is simple interest?

interest that is only paid on the principal (note: to find the final amount, add the interest in dollars to the starting amount, or use the formula for compound interest without n [A=P(1+rt)] because the n=0)

what is compound interest?

interest that is paid on the initial amount and on previously earned interest as it accumulates

exponential functions and logarithmic functions are ___________

inverses (because log functions are y=log𝔟(x), and exponential functions are y=bˣ, so x and y have switched spots between exponents and answers)

principal

original amount of money that is put in (deposited) or borrowed from (loan) a bank

How to find the asymptote for exponential functions?

the horizontal line will be at y=k

What is the equation for exponential functions?

y= (a)Bᵇ⁽ˣ⁻ʰ⁾+k a→ vertical stretch/shrink by a factor of a -a→ reflection over the x axis b→ horizontal stretch/shrink by a factor of 1/b -b→ reflection over the y axis h→ shift horizontally k→shift vertically y and x→ coordinates B → base of the function (B^b shows whether it is growth or decay, as long as a is positive)

What is the model for exponential growth?

y=C(1+r)ᵗ y= final amount c= original amount 1+r = growth factor (must be >1) r= growth rate (must be in decimal form) t= time period