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Asymptote

An imaginary line on a graph that acts as a boundary line.

Base

the number that is written with an exponent

Common logarithm

A logarithm whose base is 10, or just log

Exponential equation

An equation that contains one or more exponential expressions

Exponential Function

y = ab^x

Exponential regression

A statistical method used to fit an exponential model to a given data set

inverse function

Let f be a one-to-one function with domain A and range B. Then its ________________ f^-1 has domain B and range A and is defined by f^-1 (y) = x, f(x)=y for any y in B

Logarithm

The exponent that a specified base must be raised to in order to get a certain value

Logarithmic equation

An equation that contains a logarithm of a variable

Logarithmic regression

A statistical method used to fit a logarithmic model to a given data set

Natural logarithm

The logarithm with base e is called the natural logarithm and is denoted by "ln"

Natural logarithmic function

The function f(x)=ln x

change to exponential form

How to solve a log equation

Logb 1

= 0

X^0

= 1

Logs with "no base"

*log with no base is always base 10

When do you simplify a log?

*fully solve the log whenever you can

No solution log functions

*a positive base with a negative log always= no solution

*always put parenthesis around what you write/ do first

How to write a long when you have both (÷) and (X) or (+) and (—)

*power/root

When/ how to convert from a radical in a log to a fractional exponent

*when log is expanded, exponent goes in front of log

Where is an exponent written in expanded log form, and where it is written in a single log?

Converting from log form to exponential

*in log equations, you are trying to find the exponent

y= #logb X

Graphing logs: compression/ stretch

Graphing logs: horizontal shift

y= logb (X - #)

Graphing logs: vertical shift

y= logb X + # OR y= logb (X-#) + #

parent graph of log functions:

What is the domain/ range of parent graph of log function, and how do you find domain/range if they change?

*only changes when there's a horizontal shift

When does the Domain of a log function change?

*only changes when there's a vertical shift

When does the Range of a log function change?

For exponentials

Exponential Growth vs. Exponential Decay

Domain and Range of Exponentials

*unless there is a transformation*

parent graph:

Transformations with exponential functions

y intercept of an exponential for parent graph

*unless there is a transformation y intercept= (0,1)

y= ab^-x

*not the same as negative in front of whole function (reflection across x axis)

How to get the value of x in exponentials using the graph

•if the function= #, find what value x is when that # is y

a=c

If b^a= b^c

Change of base

logb X= log (x)/ log (b)

e

*continuous growth factor≈ 2.718

Base e

*base e= ln

Solve exponentials w/e

*to solve exponentials with e, take the ln of both side

y= lnx

Inverse of y= e^x

Compounded Continuously Formula

A= P • e^rt

Compounded Annually Formula

y= a(1+r) ^t

10^m1-m2

Earth Quake Intensity

if g(x) is the inverse of the function f(x)

the f(g(x)) is equal to x and g(f(x)) is equal to x

f^-1(x)

to denote the inverse function of f(x) write

to determine if a function has an inverse or is invertable

use a horizontal line test. For a function to have an inverse any y value must only map to one x-value.

one to one function

a function that has an inverse that is also a function

Horizontal Line Test

inverse of a function is also a function if and only if no horizontal line intersects more than one

Graphically two functions are inverse of each other if

they are a reflection of each other over the y = x line

Algebraically two functions are inverses of each other if

f o f-1(x) = x and f-1 o f(x) = x

f composed of f inverse

f o f-1(x) is read as

first identify several points of the graph f,

To reflect the graph of a function f across the y = x line,

To find the inverse of a function

swap the x and y values and solve for y, if the new equation can be solved for y as a function of x, it is the inverse of the original function. (even functions are not invertable)