5.1-5.6
What is ln1
0
what is log(1)
0
What to know about base e
1. e is just a number (kinda like pi) 2. e and natural logs are inverses of each other 3. e: (1+(1/n))^n as n->infinity When this is done out in a graph, e approaches the limit as 2.71828, which is what it is = to
What is the common log base (i.e. if no base (subscript) is written, what is the base
10
How to solve an equation with 2 logs on one side of the equation with 2 different methods where both logs have the same base and one side of the equation =0
1: go long form -> short form, left right left, cross multiply if applicable, simplify 2: move one log to the other side of the equation so there is one log per side, divide by LOGb, simplify
Exponential graphs
2^x is a positive slope up and right (growth) 2^-x is a negative slope up and to the left (decay) -2^x is a negative slope down and to the right
How many decimal places do logs go to
4
Ex solve 4LOG81(x+5)-1=2 (81 is the base)
4LOG81=3 LOG81(x+5)=3/4 81^(3/4)=x+5 (right left right) (^4√81)^3=x+5 27=x+5 22.0000=x Remember to plug the value for d back in to make sure that the domain is still positive (not - or 0)
Compound Interest Formula
A=P(1+r/n)^nt A- amount + interest P- principle (starting amount) r- rate as a decimal n-how often interest is accrued per year (so like every month would be 12) t-time (years)
continuously compounded interest formula
A=Pe^rt A- amount + interest P- principle (starting amount) r- rate as a decimal t- time (years) When solving using this, remember e is a number , not a variable that will be changed with a number
How to go long form-> short form
Bring any coefficients up as exponents (if the coefficients are fractions, then make them roots), make anything added multiplied and anything divided subtracted with one log sign
How to solve a function where e^# =e^#•e^#
Change the left side to be e^#+# Use one to one property to solve by setting the exponents equal to each other Factor and solve
How to solve a log when after doing left right left, the side with the variable is a fraction
Cross multiple and simplify
How to find the domain, vertical asymptote, and x-intercept given ln
Domain- set the argument >0 and solve. Final answer will be in interval notation (one side being an infinity) Vertical Asymptote- Set argument =0 and solve X intercept- Method 1: (left-right-left) 1. change ln to loge 2.Then set =0 3. Left right left 4. Simplify to solve for x (write final answer as a point) Method 2: (exponentiate) 1. Set =0 2.Make both sides the exponent of an e 3. Simplify and solve writing final answer as a point (Remember e^ln=1 and e^0=1)
Properties of y=lnx
Domain: x>0 Range: all reals Increasing Concave down ln(x1•x2)=lnx1+lnx2 ln(x1/x2)=lnx1-lnx2 ln(x2)^(x1)=x1lnx2 Vertical asymptote x=0 lne^x=x
Properties of y=e^x
Domain:all reals Range:y>0 Concave up Increasing One to one function e^x1•e^x2=e^(x1+x2) e^(x1/x2)=e^(x1-x2) Horizontal asymptote y=0 e^lnx=x
Which logs can have variables=-#
Equations where the variable is not part of the domain, equations where the variable is an exponent, or equations where even with a negative variable the domain wi still be positive.
How to solve an equation using one to one property when a fraction ^x could only possibly have the same base as the reciprocal of the fraction it is equal to
Flip the fraction ^x and make it ^-x and then make the bases the same by finding the exponents. This can be placed outside of the whole fraction to make the whole fraction ^-#x (if the numerator of the fraction is 1, the exponent from the denominator can still be pulled out to be for the whole fraction because 1^any # = 1. Set exponents equal once the bases are the same and solve
How to go from short form to long form
Follow all of the properties of logarithms, so change any products being multiplied to be logs added together and any being divided to be a log subtracted from the others. Additionally, if there are any exponents, bring them in front
How to find half life
Graph the exponential function and half of the original sample (just y=#) and find the point of intersection using the calculation
One to one property
If a^x=a^y, then x=y (so if there is a common base, then exponents are equal)
How to solve equations with one to one property with decimals related to 10
It is like scientific notation like 1•10^x so the decimal will be 10^-# with the number of times the decimal was moved to the left from 1. Make the other side have 10 as a base and solve like normal.
Quotient property of logs
LOGb(M/N) = LOGbM-LOGbN
Product property of logs
LOGb(MN) = LOGbM+LOGbN
Power Property of Logs
LOGbM^k = kLOGbM (down in front)
Ex solve e^2x-2e^x-63=0
Let u=e^x u^2-2u-63=0 (u-9)(u+7)=0 u=9 u=-7 e^x=9 e^x=-7 lne^x=ln9 lne^x=ln(-7) x=ln9. Extraneous x•2.1972
How to solve an exponential function
Log both sides (because a log undoes an exponential function), down in front for any exponents, divide both sides by the smallest log (excluding exponents) and simplify to solve for x *remember if the problem didn't start with log it doesn't end with it
Ex solve 3^x=20
Log3^x=log20 Divide both sides by log3 X=(log20/(log3) X=2.7268 Check this by plugging it back into the original equation
What is the inverse of exponential functions
Logarithm
How to solve using one to one property when there are numbers other than one in the numerator and denominator of a fraction on one side
Make it so the numerator and the denominator have the same base as eachother and the other side of the equation. Then for the side of the equation with the fraction move the denominator to the numerator by subtracting the exponent of the denominator from the exponent of the numerator (remember to distribute the negative). Solve like normal
How to use one to one propertty
Make sure the bases are the same and then you can set one exponent equal to the other and solve. If one side is a binomial factor by using zero product property
How to solve with one to one property when 1 is the numerator and the number with the exponent x is in the denominator
Make the exponent negative (distribute negative if the exponent is a binomial) and move the number with the exponent to the numerator. Make bases equal and solve by setting the exponents equal to each other.
How to solve using one to one property when the number ^x needs a different base
Make the number into the number ^2 or ^3, etc times x Ex 9^x=3 3^2x=3^1 2x=1 x=1/2
How to solve an equation where there is a natural log on each side of the equation and one side is a binomial (so you can't just divide by the natural log)
Move both terms with ln to one side of the equation (the side without the other term without ln) Short form-> long form Change ln to LOGe Right left right Cross multiply if applicable Simplify (May have to factor out the variable then divide) ~ final answer most likely will have an e in it except when e^0 ~ plug the number back in when the equation starts with log or ln to make sure the domain is positive
Will there be any extraneous values with answers with the one to one property
No
What should you write when going short form -> long form when there is LOGb1
Nothing because LOGb1=0
How to check to make sure you solved using one to one property correctly
Plug the value back in and make sure it works
How to solve an equation using one to one property when one side has a base with an exponent and the other side is just 1
That one equals base^0 Solve like normal
How to go short form -> long form when one of numbers being logged is a square root
That really means it is x^(1/2) so bring 1/2 down in front
How to covert an exponential function to a log
The base becomes the subscript of log, the exponent becomes what the log is equal to, and what the exponential function was equal to becomes the number next to the log Ex. 4^3=64 log4(64)=3 (4 is a subscript)
What to remember about properties of logs
They are for when M and N are positive, base >0, and base not equal to one THE PROPERTIES GO BOTH WAYS
How to find how long it takes to double the amount
Use rule 72 This is ONLY for growth, not decay 72/(rate as a %) This is how long it would take
How to write an equation given a starting amount with percent interest per year in addition to determine the amount made after a number of years
Use y=a0b^x (0 is a subscript) a0- starting amount b- growth factor ->100%+% increase -> 1+ 0.## (decimal form) x- number of years Hill Just fill these in to write the equation When solving, fill in the number of years for x and solve
How to write a function given 2 points
Use y=ab^x/y=ab^x and plug in each of the two points for each x and each corresponding y. Make sure to put the point on top that will result in a prostitute b exponent on top. Cancel whatever can be (most often a's can and b's) then take the root if applicable to find what b=. Then choose 1 point and Dillon x,y, and b in y=ab^x. Solve for a. Then write the formula by filling in the a and b in y=ab^x
How to write a function given an initial amount and percent decrease for a time period
Y=a0b^x Plug in initial amount of a0 Plug in 100%-% for b -> 1-0.## Plug in a number for x for the number of hours, years, etc
What is y=ln(x) the same as
Y=lOGe(X) where e is the base Because ln=LOGe
If there is a log with more than one domain, do both domains have to be positive
Yes, so if the variable value makes only 1 negative it still is extraneous
How to solve a function ex (e^(x+2)) ln (1-2x)=0
ZPP so you have e^(x+2)-1= 0 and ln (1-2x)=0 e^(x+2)-1= 0 e^(x+2)=1 lne^(x+2)=ln1 x+2=0 x=-2 And ln (1-2x)=0 LOGe(1-2x)=0 e^0=1-2x 1=1-2x X=0 *had to check both answers to make sure the domain for ln was positive
Ex evaluate log6(11)+log4(13)
[log11/log6]+[log13/log4] Then put this in your calculator but pay attention to parentheses. Put in [log11/log6] and then press sto -> , enter. Then do [log13/log4] + x Answer= 3.1885
Exponential function
a0(b^x) 0 is a subscript a0- initial value (initial value) b- growth or decay x- time period
Asymptotes of exponential functions and logs
an exponential function (y=2^x) has a horizontal asymptote y=logx has a vertical asymptote
How to evaluate when given a log with no =
assume it is equal to y Use left right left to make it into an exponential function evaluate (find what y=) by using the one to one property
Ex. How to evaluate logb(1/64)=-6
b^(-6)=(1/64) b^(-6)(-1/6)=(1/64)^(-1/6) b=64^(1/6) b=(6√64)^1 b=2
What is special about natural logs
e^(ln(x))=x ln(e^x)=x So when these are present, you can set up x(exponent)=x(right side of original equation)
What is log (-#)
error (not possible)
What is log(0)
error (not possible)
How does the relationship between exponential functions and log relate to other inverse relationships
it is similar to how y=b^x is x=b^y
How to convert a log to an exponential function
left right left The subscript of log becomes the base where =what the log was equal to is the exponent. The number that was next to the log (other than the subscript) becomes what the exponential function is equal to.
Ex solve e^(2t-1)=5
ln(e^(2t-1))=ln5 ~ so natural log both sides 2t-1=ln52t=ln5+1 t=(ln5+1)/2 t=1.3047
What is change of base
logb(M) =(logm)/(logb)
Natural log function
y=ln(x)
What is the function equation of a log
y=logb(x) reads base b log of x
