CH. 4 Logs and Exps.

Logs

100

Expand: Log((x³y⁴)/z⁶)

3logx + 4logy - 6logz

200

Combine: ln(a + b) + ln(a - b) - 2lnc

Ln(a² - b²)/c²

300

Solve for x: 50/(1 + e^-x) = 4 (round to 4 decimal places)

x = -2.4423

400

Solve for x: log₂(x²-x-2) = 2

x = 3

500

A googol is 10¹⁰⁰ and a googolplex is 10^googol. Find log(log(googol)) and log(log(log(googolplex)))

Both answers are 2 

Exponents

100

Write the exponential form of the equation log₈8 = 1

8¹=8

200

Write the exponential form of the equation log₈^(⅛) = -1

8^-1=1/8

300

Solve the equation 3^x+2 = 7 algebraically

-1.43

400

Solve for x in the equation e^2x=16

1.39

500

Graph the function and state the domain, asymptotes and range: f(x)=3^(x+1)+2

domain: (-∞,∞), range: (2,∞), asymptote: HA y=2

Wild Card

100

Graph 3^-x (without a calculator)

(Goes through points (-2, 9), (-1, 3), (0, 1), (1, ⅓), and (2, 1/9)

200

Solve for x: log₃(x²+5) =  log₃(4x²-2x)

x = -1 or x = 5/3

300

Evaluate 5e^4ln(x+3) when x= -1

80

400

How long will it take $1,200 to triple if invested at 6% interest compounded monthly?

18.36 years

500

The number N of bacteria in a culture is given by N = 200e^kt. If N = 300 when t = 4 hours, find k to four decimal places and then determine how long it will take for the number of bacteria to triple in size.

k=0.1014, t= 10.8380 hours

Word Probs.

100

A sum of $1000 is invested at an interest rate of 12% per year. Find the amounts in the account after 3 years if interest is compounded monthly

P=$1000, r=0.12, t=3. Using A(t) = Pert the answer is $1430.77

200

If $3000 is invested at an interest rate of 9% per year, find the amount of the investment at the end of 5 years if it is compounded continuously.

3000e^0.45 = $4704.94

 

300

A man invests $5000 into an account that pays 8.5% interest per year, compounded quarterly. When will it double?

8.24 years 

10000=5000[1+ (0.85/4)^4t

400

The half-life of Carbon-14 (14C) is approximately 5730 years. If a sample initially contains 200g of Carbon-14, find the quantity remaining after 10,000 years

P(10000) = 200 * (½)^{(10,000/5730)} = 59.66

500

The population of Atlanta is currently 50,000 and it is growing at a rate of 5% per year. The population of New York City is currently 80,000 and it is growing at a rate of 3% per year. In how many years will the population of Atlanta exceed the population of NYC?

50,000×(1.05)^t > 80,000×(1.03)^t

T is greater than 9.633 (approx. 10 years)