Exponential and Logarithmic Functions

No solution log functions

*a positive base with a negative log always= no solution *if a log doesn't work or is not solvable= no solution ex. log2 (-32)→ 2^x= -32→no solution

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

*always put parenthesis around what you write/ do first ex. log 5x/ 4y= (log 5 + log X) — (log 4 +log y)

Base e

*base e= ln *logs with base e are natural logs

e

*continuous growth factor≈ 2.718 *value of e= the asymptote

When do you simplify a log?

*fully solve the log whenever you can ex. log2 4(2)/8→ log2 8/8→ log2 1→ 2^x= 1→= 0

Converting from log form to exponential

*in log equations, you are trying to find the exponent log: logb (base) C (power)= a (exponent)→ exponential: b (base) ^a (exponent)=C (power)

Logs with "no base"

*log with no base is always base 10 Ex. log 5=log10 5

y= ab^-x

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

When does the Domain of a log function change?

*only changes when there's a horizontal shift

When does the Range of a log function change?

*only changes when there's a vertical shift

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

*power/root *you can leave logs with a √, except when √ is in the denominator (never allowed), so rewrite as a fractional exponent *however if there is a fractional exponent in the numerator, then change to a radical ex. log (162^1/4) / (∜2) → log (∜162) / 2^1/4

Solve exponentials w/e

*to solve exponentials with e, take the ln of both side Ex. e^x-2=12→ x-2 ln(e)=ln(12)

y intercept of an exponential for parent graph

*unless there is a transformation y intercept= (0,1) *to find y intercept substitute 0 in for x to get a point

Domain and Range of Exponentials

*unless there is a transformation* D: all real numbers R: y > 0 *D and R or exponentials is opposite of D and R of logs

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

*when log is expanded, exponent goes in front of log *when single log, exponent left in exponential form

What is a log

1. and inverse of an exponential equation (logs and exponentials are opposites) 2. reflection across the line y=x

Solving logs or exponentials by finding a common base

1. convert to exponential form 2. substitute the two values (base) and (power/answer) for number broken down into a common base 3. set exponents = to each other and solve ex. log8 32 → 8^x=32 → (2^3)^x= 2^5 → 2^3x= 2^5 → 3x=5→ x= 5/3

Properties of logs

1. product → multiplication= add logs 2. quotient→ division= subtract logs 3. power→ exponents= multiplication

How to find the inverse of a log function

1. replace x and y with each other 2. switch to exponential form 3. isolate y

How to solve exponentials

1. try to find a common base 2. if you can't get a common base you must take the log of both side

Earth Quake Intensity

10^m1-m2 *m= magnitude *m1 is always the bigger number

Logb 1

= 0

X^0

= 1

Compounded Continuously Formula

A= P • e^rt A= final amount in account P= principal amount in accout r= interest rate (annually) t= time in years

Exponential Growth vs. Exponential Decay

For exponentials Growth: f(x)= 2(5)^x → # in parenthesis is greater than 1 Decay: f(x)= 3 (3/4)^x → # in parenthesis is less than 1, must be a fraction

If b^a= b^c

a=c *when bases are the same and set equal to each other, the exponents must be equal

How to solve a log equation

change to exponential form

Change of Base

logb X= log (x)/ log (b) *always log of answer over log of base

Transformations with exponential functions

parent graph: y= ab^x transformations: y= a(compression/stretch)b^x-h (horizontal)+ K (vertical) *for horizontal shifts (-)= right (+) = left *when there's a negative in front of a, then there's a reflection across the x axis

Graphing logs: compression/ stretch

y= #logb X *the compression/stretch goes in front of the log *less than 1= compression *greater than 1= stretch

Compounded Annually Formula

y= a(1+r) ^t a= initial amount r= percent increase (decimal) t= time in years *if problem says decrease than a(1-r)^t

Inverse of y= e^x

y= lnx ln= log(e)

Graphing logs: horizontal shift

y= logb (X - #) *horizontal shift always in parenthesis *shift is opposite of sign ex. (x-5)= right 5

Graphing logs: vertical shift

y= logb X + # OR y= logb (X-#) + # *vertical shift is never in parenthesis *shift is not opposite of sign ex. logb X-7= down 7

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

•if the function= #, find what value x is when that # is y ex. 2^x= 8→ when 8 is y on the graph, x=3

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

•parent graph of log functions: Domain= x>0 Range= all reals • D and R change when there are transformations, so to find D and R simply find the horizontal or vertical shifts

Asymptote

•the asymptote it the line of the graph (of an exponential or a log) almost but never reaches •for the parent graph the asymptote is always 0 (so the x axis) *asymptote only changes from 0 when there's a vertical shift