Show:
Exponentials & Graphs
Logarithms
Solving
Interest, Growth & Decay
Exponential Regression & Review
100
What happens if the exponent is negative?
Reflection over y axis.
100
What is the inverse of f(x) = a^x
f(x) = log x a
100
Solve for x: 5^(x+1) = 625 hint: get equal bases
5^(x+1)= 5^4 x+1=4 x=4-1 x=3
100
What is the compound interest formula?
A = P(1+r/n) ^nt
100
What kind of graph depicts how hot water will cool down in room temperature?
Exponential decay
200
Graph the parent exponential function.
Make sure the graph goes through (0,1) and is rapidly increasing positively.
200
Find log 34 9
log34/log9=1.6049
200
Evaluate 7 = log 2^7 2
The 2'S cancel out and 7=7
200
The amount of radioactivity, A, remaining in a laboratory sample is give by the formula A = 170e^(-.035t) where t is the number of days since the start of the experiment. How much radioactivity is left in a laboratory sample 6 days after the start of an experiment?
plug in 6 for t because t is the variable for time and 6 is the given from the equation A = 170e^(-.035(6)) 137.8
200
What variable tells you whether the equation is a good fit for the graph?
r^2
300
Complete the rules. a^0 = ? a^1 = ? a^x = ?
a^0 = 1 a^1 = a a^x = a^x
300
Graph f(x) = log (x-1) 2
Make sure you have the points (1,0) (2,1)
300
ln(ln(ln(x+6))) = 0
1=ln(ln(x+6)) e^1=ln(x+6) e^e=x+6 x = e^e - 6
300
A colony of bacteria doubles in 3 hours a) Find a function that models the number of cells in the culture. b) How long will it take for the colony to triple?
a) A= A e^kt 0 k = (ln2)/3 b) 4. 77775 hours
300
e^(3x-1) = 1 _________ e^(2x+4)
e^(3x-1) x e^(2x+4) = 1 e^(5x+3) = 1 ln e^(5x+3) = ln 1 5x+3 = 0 5x = -3 x = -3/5
400
What logarithm function is also an exponential function?
Natural log (e^x)
400
List four properties of Logs
log a + log b= log (ab) m m m log a - log b= log (a/b) m m m log a^n = nlog a m m log b = logb/loga a
400
log(10x+5) - log(x-4) = log2
Log(10x+5/x-4)=log2 10x+5/x-4=2 2(x-4)=10x+5 2x-8=10x+5 8x=-13 x=-13/8 DNE, can't take log of a negative number
400
You decide to invest $1400 for 2 years into a saving account that pays 4.5% interest. How much money will you have after two years if you invest it into an account that compounds interest use Compound Interest Formula a) yearly b) monthly c) daily d) continuously
a) A= 1400(1+ .045/1)^1(2) A= $1528.84 b) A= 1400(1+ .045/12)^12(2) A= $1531.59 c) A= 1400(1+ .045/360)^360(2) A= $1531.84 d) A= Pe^rt A= 1400e^.045(2) A= $1531.84
400
Solve y=2^(-x) - 3 a) the y intercept b) the shift of (0,1) c) the horizontal asymptote
a) plug in 0 for x (0,-2) b) they y value is 1, so you shift the y intercept up 1 coordinate ---> (0,-2) c) it is -3 because the function is being shifted down 3 from the function y = 2^(-x) ---> y =-3
500
y = 3^(x-1) - 2 For the problem identify: a. the y-intercept b. the additional point (the shift of (0,1)) c. the horizontal asymptote
a. plug in 0 for x (0, 5/3) b. it is subtracting 1 from 0 (x)(1, -1) c. it is -2 because it is shifting the function down y= -2
500
Solve: 4^3x = 5^(x+1)
log 4^3x= log 5^(x+1) 3x log4 = (x+1)log5 3x log4 = x log5+ log5 3x log4 - x log5 = log5 x(3log4 - log5) = log 5 x= log5 -------------- (3log4 - log5)
500
solve 2^(4x-1) = 3^(1-x)
log2^(4x-1) = log3^(1-x) (4x-1)log2 = (1-x)log3 4x log2 - log2 = log3 - x log3 4x log2 + x log3 = log3 + log2 x(4log2 + log3) = log3 + log2 x= log3 + log2 -------------------- (4log2 + log3)
500
Suppose your parents have agree to sell you their car in two years for $3500. Your savings account still pays 4.5% interest, compounded monthly. How much money will you have to invest now in order to have $3500 at the end of the two year period?
A= P(1+ r/n)^nt P= A(1+ r/n)^-nt P= 3500(1+ .045/12)^-12(2) P= $3199.30
500
Solve for X: log (3x-5) = 5 4
4^5 = 3x-5 1,024 = 3x-5 1,024+5 = 3x 1,029 = 3x x = 343