4: exponential and log functions
growth decay/rate for a different time period formula
((1 + or - r)^ new time/original time)^time period where everything is in the same units
LOG b 1 -->
0
LOG b B -->
1
Ln e =
1
anything raised to the zero exponent is __
1
number e
2.718, represents continuous growth or decay
half life formula
A=Ao(1/2)^t/h where a= end amount, Ao= starting amount, t= time, h= half life
compound interest formula
A=P(1+r/n)^nt where a is end amount, p is starting amount, r is rate, t is time, and n is # of times compounded per year
continous compound interest
A=Pe^rt where p is starting amount, r is rate, and t is time
Change of base formula for logs
Base goes to bottom: Logb (~~~) = log(~~~)/logb; ex. Log3 (x+1)—> log (x+1)/log3
Domain & range of a logaritihm
Domain is x>0 because it never touches y axis; range is -infinity to infinity
Graph of a logarithim
Its inverse is an exponential function so if you have that equation, flip points which reflects over line y=x
LOG b (X * Y) with log laws -->
LOG b X + LOG b Y (product=add logs)
LOG b (X / Y) with log laws -->
LOG b X - LOG b Y (quotient=subtract logs)
How do you determine the domain for logs?
Set x or numbers in parenthesis to greater than 0 then solve. If dividing by a negative, flip > sign. Ex. Log7(x+3) set x+3>0 then -3 on both sides and you get domain is x> -3
LOG b (X^ Y) with log laws -->
Y * LOG b X (exponent=goes to front of log)
when you muitply two numbers with the same base you __ exponents
add
exponential growth function
base (b) is greater than 1
exponential decay function
base (b) is less than 1
how do you find the answer the a number raised to a fraction exponent?
bottom number of the fraction is what root the number is and the top number is what the number inside the square root is raised to (ex. 125^1/3 --> ∛125^1 --> 5)
equation for exponential functions
f(x)=ab^x where a is the y intercept & b is the growth or decay factor
if given a key (ex. LOG 7=k) and another log (ex. LOG(4900)) and you have to solve in terms of the variables, what do you do?
first think of how you can rewrite 4900 in terms of 7 then write it as LOG b of 7^x (if there were two equations in the key figure out how you can rewrite number in terms of both numbers) use log laws and you end up with the equations in the key, probably with a coefficient.
how do you use half life formula?
if you're given half life and time its been decaying: put in the t/h, make sure units match and you will get what fraction is left after that time (ex. half life is 5 seconds & how much is left in 1 minute --> t/h=60/5)
how do you write an exponential growth formula?
in y=ab^x, starting value is a and b is 1 + the percent in decimal form
how do you write an exponential decay formula?
in y=ab^x, starting value is a and b is 1 - the percent in decimal form
logarithim
inverse of an exponential function in the same base
the number e
irrational #, used in exponential modeling especially when a problem involves continuous growth or decay
asymptote
line that the function approaches but never crosses
Common logs
logs with base 10, are so common you can just write as logx instead of log10(x)
negative exponent means
move the base to the opposite side of the fraction and make exponent positive
exponents can be distributed over ___ & ___ but not __ & __
multiplication & divison but not addition or subtraction
when you muitply a power to a power you __ exponents
multiply
log base e =
natural log (ln)
how do you solve for an variable in exponent with number e in the equation?
once you're at a point where you can't solve it, natural log both sides instead of normal log. (add ln to both sides)
how do you use the compound interest forumla?
plug in percent increase as decimal and divide this number by how many times the year is broken up (ex. quarterly=4, monthly=12, daily= 365) and in the exponent you put the time frame given multiplied by how many times the year is broken up by
how do you use continuous compound interest formula?
plug in values and if you're solving, you can natural log both sides because you have number e.
how do you find the effective annual rate?
plug in values and when you get e^r, set it equal to 1+r and solve for percent OR if its normal compound interest, use a=p(1+r/n)^nt but just plug in 1+r/n & exponent n and solve then set equal to 1 +r and solve
how do you determine growth/decay percent for a different time period?
plug in values for t/h then with this answer: if it is greater than one, subtract 1. then set this equal to r and thats the answer; but if it is less than one, subtract 1. then set this equal to -r and you divide by -1 and thats the answer
how do you determine growth/decay rate for a different time period?
plug in your starting value, plug in the new tim over the old time which is converted to the new time (if the old is one year and you're trying to get to one month, you do 1/12); for the time period you multiply that denominator times variable (t)
how do you determine the equation of an exponential function?
plug the two points into the y=ab^x form. divide the two equations and put the one with the larger exponent on top so that you get a positive exponent. As cancel out and solve for b by using a root to cancel out the exponent. plug in this b value and solve for a
exponents
represent repeated multiplication
when you divide two numbers with the same base you __ exponents
subtract
how do you convert from exponent to log?
the output of a log is an exponent, ex. 2^3=8 so number that has an exponent is the base and the answer is x so log₂8
how do you convert from log to exponent?
the output of a log is an exponent, ex. log₂8 so you think 2 raised to what power would give you 8, 3 which is the answer. sometimes you may need to find a common base to help
what does a fraction exponent mean?
the same as a square/cube/etc. root
half life
the time required for a substance to decrease to half its original amount
how do you you logarithm approach instead of using method of common bases?
turn both variables on both sides into logs, all you do is add the word log to the front (ex. 4^x=8 --> log 4^x =log 8) use log laws and solve to get x alone
method of common bases
utilized in an equation when you have to solve for an x in an exponent. find a number that can be raised to a certain power to reach both numbers on both sides of the equal sign. multiply the exponents by the original exponent value in the original equation. since the bases are the same you can just set the two exponent values equal to each other and solve for x
LOG b B^X -->
x because b^? = b^x and same base = same exponent
