Chapter 7 Review Alpha

Basics: Exponentials

100

The function f(x)=0.8(1.7)^x represents growth or decay and why?

Growth, because 1.7 > 1.

200

The function f(x)=9.3(0.6)^x represents growth or decay and why?

Decay, because 0 < 0.6 < 1.

"0.6 is between 0 and 1."

300

Would the function g(x)=18,000(0.75)^x represent the appreciation of a home value, or the depreciation of a car value? Why?

Depreciation; 0.75 represents a decay constant because it is between 0 and 1.

400

Sketch the graph of f(x)=2(1.3)^x with an accurate y-intercept.

See board.

500

Sketch the graph of f(x)=-4(0.12)^x, including an accurate y-intercept.

See board.

Basics: Logarithms

100

What is the base in the expression log(100)?

10; this is the common logarithm.

200

What is the base in the expression ln(20)?

e, which is about 2.71828. This is the natural logarithm.

300

Rewrite 3⁴=81 in logarithmic form.

log₃(81)=4

400

Rewrite log₆(1/216)=-3 in exponential form.

6^-3=1/216

500

What is the change of base formula? Give an example of its use.

Change of base formula:

log_a(b) = log(b)/log(a)

Example with calculator:

log₈(4)=log(4)/log(8)=2/3

Example without calculator:

log₈(4) = log₂(4)/log₂(8)=2/3

Writing Equations

100

You buy a car worth $50,000, but it depreciates 10% a year. Write an equation to represent the depreciation over time.

A(t)=50000(0.9)^t

200

A sample of bacteria in a petri dish begins with 120 individuals. The bacteria double themselves every hour. Write an equation to represent this.

B(t)=120(2)^t

300

Your cashier gives you back a rare coin in your change. It was worth 25 cents in 1909, and it increased in value (on average) 6.8% each year. What is its value now?

The coin is worth approximately $347.36 in 2019.

Explanation: Let t=years since 1909. Then t=110.

A(t)=0.25(1.068)^t

A(110)=0.25(1.068)¹¹⁰=347.358399...

400

Some bacteria in a petri dish begin with 60 individuals. They reproduce by doubling every half hour. How many bacteria are present after 8 hours?

There are almost 4 million bacteria.

Explanation:

B(t)=60(2)^t

After 8 hours, the bacteria have split 16 times, so let t=16. B(16)=60(2)¹⁶=3,932,160.

500

You've discovered a new atom isotope: Stadtfeldium-256, abbreviated St-256. You begin with 90 grams of St-256, but it decays at a rate of 40% per minute. If it takes you 5 minutes to run across the laboratory to show your peers, how many grams of St-256 is left by the time you get there?

Approximately 7 grams remain.

Explanation:

A(t)=90(0.6)⁵=6.9984

Changing Forms

100

Rewrite log₂(8)=3 in exponential form.

2³=8

200

Rewrite 11²=121 in logarithm form.

log₁₁(121)=2

300

Rewrite log(1)=0 in exponential form.

10⁰=1

400

Rewrite ln(403.428793...)=6 in exponential form.

e⁶=403.428793...

500

Rewrite 5log(x)-log(x+2) as a single logarithm.

log(x⁵/(x+2))

Evaluating Logs

100

Calculator: Evaluate log(0.12). Round to four decimal places.

-0.9208

200

100 points each:

Calculator: Evaluate ln(8). Round to four decimal places.

No calculator: Evaluate log₁₄₀₈₉(1).

ln(8) is approximately 2.0794.

log₄₈₉(1)=0 because 14089⁰=1.

300

No calculator (100 pts each):

Evaluate log₃(27).

Evaluate log₄(1/16).

Evaluate log(0.1).

log₃(27)=3 because 3³=27.

log₄(1/16)=-2 because 4^-2=1/16.

log(0.1)=-1 because 0.1 is 1/10, and 10^-1=1/10.

400

200 points each:

Evaluate log₁₁(26) using a calculator. Round to four decimal places.

Evaluate log₇(900) using a calculator. Round to four decimal places.

log(26)/log(11)

1.3587

log(900)/log(7)

3.4957

500

No calculator:

Evaluate log₃₂(4).

log₂(4)/log₂(32)=2/5