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
