*Writing Exponential Equations: Do an exponential regression
*Writing Exponential Equations: Do an exponential regression
y = a(b)x a=y-intercept b=starting value. Growth if b>1
*Writing Equations in Log Form:
Base of exponent becomes Base of log. The isolated number goes with the log
Example: 52 = 25 ( 2 = log5 25
If you have an e then you just put “ln’s” to cancel: e5 = x ( 5 = ln x
If you have an ln then you just put “e’s” to cancel: ln y = a ( y = ea
*Typing logs into calculador
Logwa = log(a) / log(w) Example: log46 = log (6) / log (4) = .7737
Log3x = log3 (x + 5)( Y1=log(x)/log(3) y2 = log(x+5) / log (3)
*Writing One Equivalent Log
Log3a + log3 2 ( log3 (2a) 2log3a ( log3 (a2) 5log3a – log3x ( log3 (a5/x)
Ex: 3log2x – 4log22 + ½ log29 ( log2x3 – log224 + log291/2 ( log2 [pic]
*Exponential Growth Problems: a is the starting value, y is the finished value
Appreciation/Growth y=a(1+r)t r is the rate as a decimal
Depreciation/Decay y=a(1- r)t r is the rate as a decimal
Other Exponential y=abx b is what is being multiplied
y=3(1.067)x implies that it STARTS at 3 and GROWS BY 6.7% (106.7-100=6.7%)
y=5(.92)x implies that it STARTS at 5 and Decays by 8% (100-92=8%)
If you are given data, then do a “regression” to find the equation.
Remember: Double something: a=1, y=2
1) Express 161/2 = 4 in logarithmic form
A. log416 = ½ B. log164 = ½ C. log1/216 = 4 D. log41/2 = 16
2) Express log2x + 4log2y as a single logarithm
A. log2xy4 B. log2x + y4 C. log2x + 4y D. log24xy
3) Express ln 20 = 3 in exponential form
A. 203 = e B. 320 = e C. e20 = 3 D. e3 = 20
4) Solve log8x = 2.3
A. 1.1 B. 2.5 C. 119.4 D. 783.4
5) Solve 4y-1/2 = 16
A. -3.46 B. 0.0625 C. 0.5 D. 64
6) Solve ex-3 = 455
A. 3.1 B. 7.7 C. 9.1 D. 22.7
7) Jack deposited $1000 at 4% interest compounded quarterly. How long will it take his money to double? (y=a(1 + r/n)nt n=# times per year the money is compounded)
A. 5 years B. 15 years C. 17 years D. 174 years
8) The number of bacteria is represented by the function A(t) = 1,000(10)t/48. About how long will it take for there to be 1,155 bacteria?
A. 3 hours B. 7 hours C. 45 hours D. 105 hours
9) Toon County had a population growth rate of approximately 1.7% from 2000 to 2001. If the population was 118,700 in 2000 and this growth rate continues, about how many people will reside in 2006?
A. 120,700 B. 124,400 C. 130,800 D. 131,300
10) A new car costs $16,750 in 2000 and depreciates 12% every year. Which equation models the value of the car after 2000?
A. V(x)=16750(0.12)x B. V(x) = 16,750(.88)x C. V(x) = 16,750(1.012)x D.V(x) = 16750(.88x)
11) If $8,000 is invested at an annual interest rate of 4% compounded continuously, how much is the investment worth in 6 years?
A. $9,920 B. $10,123 C. $10,170 D. $49,958.92
12) f(x) = 200(1.03)x. What does the 200 represent? What doe the .3 represent?
A. Starts at 1.03 and increases by 200% B. Starts at 200 and increases by 30%
C. Starts at 200 and increases by 103% D. Starts at 200 and increases by 3%
13) The table below shows the population of Smithville. Which exponential equation best models the data? (t=0 in 1965)
|Year |1965 |1976 |1984 |1990 |1995 |
|Population (thousands) |50 |93 |146 |205 |271 |
A. P(t) = 52(1.53)t B. P(t)=52(1.053)t C. y=6.044t + 40.24 D. y = 50(1.057)t
14) A room at Motel 8 was $8 in 1950. It cost $49.21 in 2000. What is the average rate of growth?
A. 37% B. 3.7% C. 8% D. 21%
15) During an experiment on bacteria growth, there were 56 bacteria 3 days after the experiment began and there were 124 bacteria after 5 days. Write the exponential equation that models the data and indicate how many bacteria there were when the experiment began?
A. y = 7,298(.2)x; 7298 B. y=17.42(1.48)x; 17 C. y = 7,298(.2)x; 2 D. y=17.42(1.48)x; 1
16) In 2000, a North Carolina population model was y=12(1.03)t-2000 million people where t is the year. What did the model predict the population to be in 2010?
A. 7,620,000 B. 16,127 C. 16,127,000 D. 22,371,600
17) Slipper production increased at a rate of about 5.3% a year between 1980 and 1988. In 1988, the production was about 3 million. If this trend continues, which equation best represents the slipper production (in millions) since 1988? (Let x = 0 in 1988)
A. y = 3,000,000(1.053)x B.y = 3(1.53)x C. y = 3(.947)x D. y = 3(1.053)x
18) Which equation is equivalent to ln 4 + 2 ln x = 5 ln 2 – ½ ln 16
A. ln (4x2)= ln 4 B. ln(4 + 2x) = ln (2) C. ln (4x2) = ln (8) D. ln(8x)=ln(2)
19) A single bacteria divides into 4 organisms every 5 days. Use the formula N(t) = No(4)t/5. No is the number of organisms at the beginning; N(t) is the number of organisms after t days. Approximately, how long should 6 bacterium be kept in a container to have about 100,000 bacterium?
A. 35 days B. 8 days C. 52 days D. 14 days
20) John has just started a job that pays a salary of $30,000. He gets 2.3% raise every increase. How much will he earn after 6 increases?
A. $32,321 B. $41,400 C. $43,400 D. $34,385
21) Solve: log4(x + 6) + log4 x = 2
A. {2, -8} B. {2} C. {[pic]} D. {6.31}
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