Chapter 4 3 Richter Scale Which 17 Less Than The

185)

R= log I

I0 gives the Richter Scale measurement of an earthquake with intensity I where I0

is the intensity of a zero–level earthquake. Make a graph which converts the intensity of

the earthquake as compared to a zero–level earthquake to the Richter Scale measurement.

185)

186)

Find x: ln x= – 2

186)

187)

Express log4 64 = 3 in exponential form.

187)

188)

Solve for t: 300 = 500(1 –e0.2t). Assume that ln(0.4) = – 0.9.

188)

189)

Find x: log 100,000 =x

189)

41

190)

A trust fund is being set up by a single payment so that at the end of 30 years there will be

$20,000 in the fund. If the interest rate is 8% compounded quarterly, how much money

should be paid initially into the trust fund?

190)

191)

Solve for x: alogax+loga4= 8

191)

Explanation:

192)

Your friend started a savings plan with 2 pennies and each day he doubled the amount

saved. Later you started a savings plan with 4 pennies and each day you quadrupled the

amount saved. Your friend has been saving for 6 less than 3 times as many days as you. In

two days you will put the same amount in savings as your friend. How many days have

you been saving?

192)

Explanation:

193)

For the function f(x) =log4x,

(a) what is the domain of f?

(b) What is the range of f?

193)

Explanation:

42

Explanation:

194)

Use the change base formula and your graphing calculator to graph

f(x) =log3x

194)

195)

Express 10–3=1

1000 in logarithmic form.

195)

196)

Solve for x: ln(x+ 1) – ln x= ln 2

196)

43

197)

A certain medicine reduces the bacteria present by 25% each day. Currently 28,000 bacteria

are present. Make a table of values for the number of bacteria present each day for 0 to 4

days. For each day write an expression for the number of bacteria as a product of 28,000

and a power of 0.75. Use the expressions to make an entry in your table for the number of

bacteria after t days. write a function N for the number of bacteria after t days.

197)

198)

Graph the population for a city with 80,000 people as a function of years if the growth is

modeled by P= 80,000e0.04t.

198)

199)

The demand function for a product is p= 60 3–q/15 where q is the number of units and p is

the price of one unit. At what price will the demand be 15 units? How many units will be

demanded if the price is $41.60?

199)

44

200)

The number of bacteria in a culture is doubling every hour. Currently the culture has 128

bacteria. Make a table of values for the number of bacteria present each hour for 0 to 4

hours. For each hour write an expression for the number of bacteria as a product of 128 and

a power of 2. Use the expressions to make an entry in your table for the number of bacteria

after t hours. Write a function N for the number of bacteria after t hours.

200)

201)

The demand function for a product is p= 180 3–q/15 where q is the number of units and p

is the price of one unit. At what price will the demand be 15 units? How many units will be

demanded if the price is $24.91?

201)

202)

Evaluate and simplify: log6 6

202)

203)

Solve for x: eln(3x+4) = 10

203)

204)

Find x: log446=x

204)

205)

What power of 9 is 36?

205)

206)

If p=320–q, by using common logarithms express q in terms of p.

206)

45

207)

The number of yearly visitors to a resort has been shrinking at the rate of 6%. Currently the

resort gets 120,000 tourists each year.

(a) If this rate continues, how many tourists will they get in 15 years?

(b) Use a graphing calculator to predict the number of years until the number of tourists

will be less than 20,000.

207)

208)

Graph f(x) =ex– 2

208)

209)

Given ln x= 7.1; ln y= 8.2, find ln x3

y7.

209)

46

210)

The population of a city is given by P= 10,000(1.04)t where t is the number of years after

1988. Find the population in

(a) 1988

(b) 1989

210)

211)

Suppose $5000 is deposited in a savings account that earns 10% compounded

semiannually. What is the value of the account at the end of 6 years? Assume no other

deposits or withdrawals.

211)

212)

The number of years it takes for an amount which is invested at an annual rate of 7% and

compounded continuously to become m times as large is a function of the enlargement

factor given t(m) =ln(m)

0.07 . Use a graphics calculator to find how many times larger (to the

nearest whole number increment) the investment will be in 10, 20, 30, and 40 years.

212)

213)

Express 2 ln 3 – ln 4 as a single logarithm.

213)

214)

Solve for x: ln e2x=x

214)

215)

If $400 is invested for 2 years at 6% compounded semiannually, find

(a) the compound amount and

(b) the compound interest.

215)

216)

Write the following in terms of ln x, ln(x– 3), and ln(x+ 1): ln x(x–3)2

x+ 1

216)

47

217)

Find x and express your answer in terms of common logarithms: 102x–3= 4

217)

218)

Find x: ln x= 1

218)

219)

Assume that log 5 = 0.6690. Determine the value of log 35.

219)

220)

An earthquake which is 520,000 times as intense as a zero–level earthquake has a

magnitude on the Richter Scale which is 2.7 more than the intensity of another earthquake.

What is the intensity of the other earthquake?

220)

221)

An earthquake which is 48,000 times as intense as a zero–level earthquake has a

magnitude on the Richter Scale which is 1.7 less than the intensity of another earthquake.

What is the intensity of the other earthquake?

221)

222)

The work done by 1 kilogram sample of nitrogen as its volume changes from an initial

value Vi to a final value of Vf during a constant temperature process is given by W= 3.2×

102ln Vf

Vi. If such a sample expands from a volume of 3 liters to a volume of 5 liters,

determine the work done by the gas.

222)

223)

A town of 1400 is growing at the rate of 8% per year. If this rate continues, what will the

population of this town be in 20 years?

223)

224)

Evaluate and simplify: ln e2+ ln 1

224)

225)

Find x: log2x= – 3

225)

48

226)

Assume that log 4 = 0.6021. Determine the value of log 1

16 .

226)

227)

Assume that log 5 = 0.6690 and log 6 = 0.7782. Determine the value of log 30.

227)

228)

The population of India was 651 million in 1980 and has been growing at a rate of 2% per

year. The population t years later is approximated by N(t) =651e.02t. Estimate the

population in India in the year 2010.

228)

229)

Find x if 7e3x= 1

229)

230)

Express 1 + ln x as a single logarithm.

230)

231)

Write the following in terms of ln x and ln(x+ 1): ln x4

(x+ 1)3

231)

232)

Suppose $2000 is invested at 6.5% compounded monthly.

(a) Find the value of the investment after 5 years.

(b) Find the value of the interest which was earned over the first 5 years.

232)

233)

The magnitude (Richter Scale) of an earthquake is given by R= log I

I0 where I is the

intensity of the earthquake and I0 is the intensity of a zero–level reference earthquake. I

I0

represents how many times greater the earthquake is than the reference earthquake. Find

the magnitude of an earthquake that is 2,000,000 times the intensity of a zero–level

earthquake.

233)

49

234)

Use the change base formula and your graphing calculator to graph

f(x) = – 3log2 (x + 4) + 1

234)

235)

What is the sum of the Richter Scale measurement of an earthquake which is 250,000 times

the intensity of a zero–level earthquake and an earthquake with intensity twice the

intensity of a zero–level earthquake? Write as an expression involving logarithms.

Simplify by combining logarithms and then use a calculator to evaluate.

235)

236)

Solve for x: 4x+1=83x

236)

237)

For the function f(x) =5x,

(a) what is the domain of f?

(b) what is the range of f?

237)

50

238)

What power of 3 is 36?

238)

239)

Find x and express your answer in terms of natural logarithms: 2e3x= 6

239)

240)

Find the domain of y=log3(x+ 1)

240)

241)

Solve for x: log3(x+ 7) – 3log3 2 =log3x

241)

242)

An earthquake which is 7200 times as intense as a zero–level earthquake has a magnitude

on the Richter Scale which is 3.6 less than the intensity of another earthquake. What is the

intensity of the other earthquake?

242)

243)

Assume an investment is guaranteed to triple every decade.

(a) Make a table of the factor of increase in the investment at each decade for 0 to 3

decades. For each decade, write an expression for the factor of increase as a power of some

base.

(b) What base did you use? How does that base relate to the problem?

(c) Use your table to graph the factor of increase as a function of decades.

(d) Use your graph to estimate when the investment will have grown by a factor of 15.

243)

51

244)

The multiplicative increase m of an investment which is invested at an annual rate of p and

compounded continuously for a time t is given by m=ept. If your annual rate is 6.75%, how

many years will it take to double your investment?

244)

245)

If an earthquake is 10x+2 times as intense as a zero–level earthquake, what is its

measurement on the Richter Scale? Write as logarithmic expression and simplify.

245)

246)

Find x: log6 36 =x

246)

52

247)

Graph f(x) =log2x

247)

248)

If the yeast has been decreasing by 80% hourly and the current amount is 1

78,125 of the

amount first measured, then the situation can be represented by 1

78,125 =1

5

t

. Represent

this equation in logarithmic form. What does t represent?

248)

249)

Solve for x: 22x– 2 ·2x– 3 = 0

249)

250)

What power of 6 is 36?

250)

53

251)

Solve for x: log x= log 3 + 2 log 4

251)

252)

Graph g(x) =1

3

x

252)

253)

Evaluate and simplify: log2 1 +log2 2

253)

254)

Assume that log 4 = 0.6021. Determine the value of log 400.

254)

255)

Find x: log2(x+ 4) = 3

255)

54

256)

Graph f(x) =2x

256)

257)

The amount of plastic being recycled increases by 30% every year. Write a function for the

factor of increase in plastic recycling as a function of years. Use a graphing calculator to

graph your function. Use the graph to estimate when the amount of recycling will triple.

257)

258)

Assume that log 3 = 0.4771 and log 4 = 0.6021. Determine the value of log 4

3.

258)

55

Answer Key

Testname: C4