Exponential Functions

b [A(t)= a x b^t]

(1 + r) or (1 - r) depending on growth or decay

ANYTHING TO THE 0

1

Horizontal Asymptote (Exponent Graphs)

A line across the y axis that is worked toward but NEVER TOUCHED OR GONE PAST; Always draw DOTTED LINE; Written as Y=#

Compound Interest Formula

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

ALWAYS WRITE PERCENT AS A DECIMAL

ALWAYS WRITE PERCENT AS A DECIMAL

A (Compound Interest Formula)

Amount in account after t years

If you have a logarithm WITH THE SAME BASE on both sides

Cancel the logs by crossing them out if they are on both sides

Solving log equations if there is only one log

Convert the log to an exponent and solve

A(t)= a x b^t

EVERYTHING IS THE SAME AS EXPONENTIAL GROWTH FORMULA BESIDES B

F(x)=ab^(x-h) + k

Exponential function formula for transformations

Exponential Growth Function

F(x)=b^x; b > 1; HORIZONTAL ASYMPTOTE established by the outside parenthesis transformation (y-axis); INFINITE BOTH WAYS ON THE X AXIS

Exponential Decay Function

F(x)=b^x; b is in between 0 and 1; HORIZONTAL ASYMPTOTE established by the outside parenthesis transformation (y axis)]; INFINITE BOTH WAYS ON THE X AXIS

Graphing Log Functions

Find the asymptote, find an easy x value to make an easy answer for the log, graph your points

Solving Log Inequalities

Find where each log is undefined, solve the actual inequality, USE ALL THE ANSWERS TO MAKE A NUMBER LINE, then test values to make a final solution

Graphing Exponential Functions

Graph the horizontal asymptote using k, Use the exponent to find the x values based on the transformation, Make a table with x and y values, and plot and point the graph

1/anything to the -1 power

IS ALWAYS WHATEVER THE DENOMINATOR IS

P (Compound Interest Formula)

Initial/Principal invested amount

r (Compound Interest Formula)

Interest rate (decimal)

Vertical Asymptote (Log Graphs)

Inverse of the exponent graphs; Written as x=#; Shifted inside the parenthesis by H; GOES UP AND DOWN

If asked to write an equation the represents an exponential situation

JUST FILL IN THE EQUATION WITH PROVIDED NUMBERS ; DO NOT DO ANY OTHER MATH AFTER

Logarithms

Just Exponents; Inverse of exponent form; X IS GREATER THAN 0 AND B IS GREATER THAN 0

When Doing Log Inequalities Where the Variable Isn't The Logicand

Just solve it normally, disregard the log inequality rules

Log Language

Log base (whatever) of x

F(x)= a (log) base (x-h) + k

Log graph formula

Finding Where the Log is Undefined (Log Inequalities)

MAKE THE LOGICAND GREATER THAN 0

Testing for Values (Log Inequalities)

Make a number line and find the areas the meet ALL THE CONDITIONS YOU HAVE SET FOR X

Expressing Things In Terms Of Common Logs

Make an equation and set whatever isn't in common log form equal to LOG BASE 10 TO THE X

Opposite

Multiply whatever it is by negative 1

Checking Log Equations

Must put your answer for the variables into logicands to make sure the total logicand is positive

n (Compound Interest Formula)

Number of times compounded per year

t (Compound Interest Formula)

Number of years

r [Exponential Decay Formula)

Percent decrease per time period; USE THE PERCENT OF THE ORIGINAL 100% SUBTRACTED PER PERIOD

Finding N Roots on a Calculator

Raise it to the 1/n power OR go to math and it's right there

H change (the number inside of the parenthesis in LOG FUNCTIONS)

Shifts the ASYMPTOTE along the X AXIS however many places REVERSED (if it is positive the graph is shifted toward negatives)

K Change (the number outside of the exponent after the base in EXPONENTIAL FUNCTIONS)

Shifts the ASYMPTOTE along the y axis however many places NORMALLY

K Change (the number outside of the log in LOG FUNCTIONS)

Shifts the function along the Y AXIS however many places NORMALLY

H Change (the exponent with x in EXPONENTIAL FUNCTIONS)

Shifts the function along the x axis however many places REVERSED (if it is positive the graph is shifted toward negatives)

Inverse

Switch the x and y; Symmetric about the line X=Y

b

The base; THE THING ACTUALLY BEING EXPONENTED; It is growth if b > 1 and decay if it is between 0 and 1 (or a fraction)

A(t) [Exponential Growth/Decay Formula)

The final amount after t time periods

If IaI is greater than 1 (Exponential Functions)

The graph is compressed

If IaI is in between 0 and 1 (Log Functions)

The graph is compressed vertically

If IaI is in between 0 and 1 (Exponential Functions)

The graph is expanded horizontally

If IaI is greater than 1 (Log Functions)

The graph is expanded vertically

If A is negative (Exponential/Log Functions)

The graph is reflected across the ASYMPTOTE

a [Exponential Growth/Decay Formula]

The initial amount

A (Exponential Functions)

The number before the base; It is multiplied to the base AFTER THE BASE IS EXPONENTED

r [Exponential Growth Formula]

The percent of increase per time period; USE THE PERCENT OF THE ORIGINAL 100% ADDED PER PERIOD

Logicand

The x of the log equation

When Writing An Exponential Function Whose Graph Passes 2 Points

USE FORMULA y=a b^x; A Y coordinate is the VALUE OF WHAT YOU HAVE AFTER X TIME, so you plug the FIRST Y VALUE in as the INITIAL AMOUNT (a), A VARIABLE as the BASE, the SECOND X VALUE as the EXPONENT(x), and the SECOND Y VALUE as the FINAL AMOUNT (what it all equals); JUST PLUG IT IN TO (Y=ab^x) where the variable "x" is the exponent

To Find NON CONTINUOUS Exponential Growth WITH CONSTANT INCREASE OVER TIME PERIOD THAT ISN'T COMPOUNDED

Use A(t)=a(1 + r)^t; Only used if given a fixed amount of time where the change happens like if it says "EVERY X" or "PER X"

To Find NON CONTINUOUS Exponential Decay WITH CONSTANT INCREASE OVER TIME THAT ISN'T COMPOUNDED

Use A(t)=a(1 - r)^t; Only used if given a fixed amount of time where the change happens like if it says "EVERY X" or "PER X"

Creating Like Bases

Use exponent/root rules to reduce each side to the same base and eliminate the base; Put the LIKE BASE INSIDE OF PARENTHESIS WITH THE EXPONENT VALUE THAT GIVES YOU THE SAME NUMBER AS THE ORIGINAL

If you are finding exponent bases and must use a negative exponent to make like bases

Use the negative exponent ON THE FRACTION TO GET LIKE BASES

Log Inequality Property

When converting logs with inequalities MAKE SURE THE X STAYS ON THE SAME SIDE OF THE INEQUALITY IT WAS ON BEFORE

Finding an Easy X Value (Log Graphs)

Write 4 log b logicands filled in so that you know make the answers easy, then find an x value for the function that makes the log have the easy answer you are looking for

If you have unequal bases on either side with the exponents unequal

You can create like bases and solve by multiplying the expoent of the like base exponent by the outer exponent; YOU TAKE THE TWO BASES AND FIND A NUMBER THAT CAN BE EXPONENTED BY DIFFERENT NUMBERS (Squared, Cubed) TO GET BOTH BASES (put then in parenthesis), THEN WRITE THAT NUMBER EXPONENTED BY WHATEVER MAKES IT THE BASE FOR EACH BASE, AND TO ELIMINATE THE BASES YOU MULTIPLY THE OUTSIDE EXPONENT BY THE INNER ONE

If you have the same base on both sides with different exponents

You can eliminate (take out) the base to have the EXPONENTS EQUAL to each other on either side of the equation sign

Log base x = y

base ^y = x