Algebra 2B

P(t) = 15,500(0.97)^t; the population will drop below 8,000 in 21.7 yr.

A city population, which was initially 15,500, has been dropping by 3% a year. Write an exponential function and graph the function. Use the graph to predict when the population will drop below 8,000.

a logarithm to the base 10

A common logarithm is ______________.

v(t) = 1000(0.85)^t; the value will fall below $100 in about 14.2 yr.

A motor scooter purchased for $1,000 depreciates at an annual rate of 15%. Write an exponential function, and graph the function. Use the graph to predict when the value will fall below $100.

f(x) = 25(0.4)x; 4 bounces.

A new softball dropped onto a hard surface from a height of 25 inches rebounds to about 2/5 the height on each successive bounce. (a) Write a function representing the rebound height for each bounce. (b) Graph the function. (c) After how many bounces would a new softball rebound less than 1 inch?

f(x) = 10(0.95)^x; about 6 units.

A quantity of insulin used to regulate sugar in the bloodstream breaks down by about 5% each minute. A body-weight adjusted dose is generally 10 units. (a) Write a function representing the amount of the dose that remains. (b) Graph the function. (c) About how much insulin remains after 10 minutes?

a line that a graph gets closer to as the value of a variable becomes extremely large or small.

Asymptote

The value of b in a function of the form f(x) = ab^x, where a and b are real numbers with a ≠ 0, b > 0, and b ≠ 1.

Base of an Exponential Function

4

Calculate log 10000 using mental math.

0.25

Calculate log^625 5 using mental math

-3

Calculate log^9 1/729 using mental math.

log^6 1296 = 4

Change 6^4 = 1296 to logarithmic form.

(27)^2/3 = 9

Change log^27 9 = (2/3) to exponential form.

11.4 yr

Clara invests $5,000 in an account that pays 6.25% interest per year. After how many years will her investment be worth $10,000?

A logarithm whose base is 10, denoted log^10, or just log.

Common Logarithm

1.3

Evaluate log^8 16.

1.5

Evaluate log^9 27.

-1.5

Evaluate: log^9 (1/27)

27.74

Evaluate: 100(0.95)^25

46.32

Evaluate: 100(1 + 0.08)^-10 Round to the nearest hundredth.

81.7

Evaluate: 100(1 - 0.02)^10. Round to the nearest tenth.

466.1

Evaluate: 100(1.08)^20

An exponential function of the the form f(x) = ab^x in which 0 < b < 1. If r is the rate of decay, then the function can be written y = a(1 - r)^t, where a is the initial amount and t is the time.

Exponential Decay

An equation that contains one or more exponential expressions.

Exponential Equation

A function of the form f(x) = ab^x, where a and b are real numbers with a ≠ 0, b > 0, and b ≠ 1.

Exponential Function

An exponential function of the form f(x) = ab^x in which b > 1. If r is the rate of growth, then the function can be written y = a(1 + r)^t, where a is the initial amount and t is the time.

Exponential Growth

log^3 27 = 3

Express as a single logarithm and simplify. log^3 108 - log^3 4

log^6 216 = 3

Express as a single logarithm and simplify. log^6 9 + log^6 24

log100000 = 5

Express as a single logarithm. Simplify, if possible. log 100 + log 1000

log100 = 2

Express as a single logarithm. Simplify, if possible. log5.4 - log0.054

log^1/3 (3) = -1

Express as a single logarithm. Simplify, if possible. log^1/3 (27)+log^1/3 (1/9)

log^3 81 = 4

Express as a single logarithm. Simplify, if possible. log^3 3 + log^3 27

log^5 3125 = 5

Express as a single logarithm. Simplify, if possible. log^5 50 + log^5 62.5

log^5(625*25) = 6

Express as a single logarithm. Simplify, if possible. log^5 625 + log^5 25

log^6 216 = 3

Express as a single logarithm. Simplify, if possible. log^6 496.8 - log^6 2.3

log^3 27 = 3

Express log^3 216 - log^3 8 as a single logarithm and simplify.

log^4 256 = 4

Express log^4 8 + log^4 32 as a single logarithm and simplify.

log^7 7 = 1

Express log^7 49 - log^7 7 as a single logarithm. Simplify, if possible.

64^2/3 = 16

Identify the exponential form of log^64 16 = 2/3

D : { x / x > 0 } R : ℝ

Identify the graph for the function f(x) = (4/5) using x-values { -2, -1, 0, 1, 2 }, the graph for its inverse, and the domain and range of the inverse.

log^7 343 = 3

Identify the logarithmic form of 7^3 = 343.

about $12,000,000

In 1626, the Dutch bought Manhattan Island, now part of New York City, for $24 worth of merchandise. Suppose that, instead, $24 had been invested in an account that paid 3.5% interest each year. Find the balance in 2008.

P(t) = 350(1.14)^t; the population will reach 20,000 in about 30.9 yr.

In 1987, the Australian humpback whale population was 350 and has increased at a rate of about 14% each year since then. Write a function to model population growth. Use a graph to predict when the population will reach 20,000.

P(t) = 6.08(1.0121)t; growth

In 2000, the world population was 6.08 billion and was increasing at a rate of 1.21% each year. Write a function for world population. Does the function represent growth or decay?

≈ 45,790

Jerome opened a library with 12,000 books in the year 1999. The number of books is increasing at a rate of 8.73% each year. Use a graph to predict the number of books in 2015.

The exponent that a specified base must be raised to in order to get a certain value.

Logarithm

An equation that contains a logarithm of a variable.

Logarithmic Equation

A function in the form of f(x) = log^b x, where b ≠ 1 and b > 0, which is the inverse of the exponential function f(x) = b^x.

Logarithmic Function

A logarithm with the base e, written as ln.

Natural Logarithm

The function f(x) = In x, which is the inverse of the natural exponential function f(x) = e^x. Domain is {x | x > 0}; range is all real numbers.

Natural Logarithmic Function

5.8 yr

On federal income tax returns, self-employed people can depreciate the value of business equipment. Suppose a computer valued at $2765 depreciates at a rate of 30% per year. Estimate the number of years it will take for the computer's value to be less than $350.

V(t) = 10000(1.1212)^t ; growth

Peter bought an antique piece of furniture in 2000 for $10,000. Experts estimate that its value will increase by 12.12% each year. Identify the function that represents its value. Does the function represent represent growth or decay?

343

Simplify 10^log343.

0.9

Simplify log10^0.9.

2/3

Simplify log^343 49.

8

Simplify log^5 625^2.

x+1

Simplify log^8 8^x+1.

-4

Simplify, if possible. log^2 (0.5)^4

x/2 + 5

Simplify, if possible. log^2 2^(x/2 + 5)

5

Simplify, if possible. log^4 1024

8x

Simplify: 2^log2(8x)

x^2

Simplify: e^ln x^2

x - 1

Simplify: log^4 4^x - 1

7/6

Simplify: log^64 128.

-2

Solve and check: 9^x = 3^x-2

1.5

Solve for x and check. Enter only a number. 3^2x = 27

108

Solve for x. log72−log(2x/3)=0

5

Solve for x. Enter only a number. 3 = log8 + 3logx.

y = 100,000(1.04)x; 2012

Some real state agents estimate that the value of a house could increase about 4% each year. (a) Write a function to model the growth in value for a house valued at $100,000. (b) Graph the function. (c) A house is valued at $100,000 in 2005. Predict the year its value will be at least $130,000.

growth

Tell whether the function shows growth or decay. Then graph.

decay

Tell whether the function shows growth or decay. Then graph. f(x) = 32(0.5^x)

decay

Tell whether the function shows growth or decay. Then graph. f(x)=0.4(3/4)^x

decay

Tell whether the function shows growth or decay. Then graph. g(x) = 2(0.2)^x

decay

Tell whether the function shows growth or decay. Then graph. j(x) = -(1.5)x

growth

Tell whether the function shows growth or decay. Then graph. p(x) = 5(1.2^x)

f(t) = 580(1.0232)t ; the number of ton-miles would have exceeded 1 trillion in year 24, or 1984.

The amount of freight transported by rail in the United States was about 580 billion ton-miles in 1960 and has been increasing at a rate of 2.32% per year since then. (a) Write a function representing the amount of freight, in billions of ton-miles, transported annually (1960 = year 0). (b) Graph the function. (c) In what year would you predict that the number of ton-miles would have exceeded or would exceed 1 trillion (1000 billion)?

2.5

The loudness L of sound in decibels is given by L = 10 log(I/1^0) where I is the intensity of sound, and I^0 is the intensity of the least audible sound. If I^0 = 10^-12 W/m^2, about how many times more intense is a 97 decibel sound than a 93 decibel sound?

N(t) = 2.5(2)^t/3

The quantity of new information stored electronically in 2002 was about 5 exabytes, or 5 x 10^18 bytes. Researchers estimate that this is double what was stored in 1999. Suppose this trend continues. Write and graph a function to predict the pattern of growth beginning in 1999.

V(t) = 3000(0.7)t; decay

The value of a $3000 computer decreases about 30% each year. Write a function for the computer's value. Does the function represent growth or decay?

V(t) = 3700(0.75)^t ; decay

The value of a Plasma TV bought new for $3700 decreases 25% each year. Identify the function for the value of the television. Does the function represent growth or decay?

≈ $899.09

The value of a piece of jewelry bought new for $2200 decreases 12% each year. Use a graph to predict the value of the jewelry in 7 years.

-2.262

Use a calculator to find log^1/3 12 to the nearest thousandth.

2.679

Use a calculator to find log^4 41 to the nearest thousandth.

D: {x | x > 0}; R: all real numbers.

Use the given x-values to graph the function. Then graph its inverse. Describe the domain and range of the inverse function. f(x) = (4/5)^x; x = -2, -1, 0, 1, 2, 3

D: {x | x > 0}; R: all real numbers.

Use the given x-values to graph the function. Then graph its inverse. Describe the domain and range of the inverse function. f(x) = 0.5^x; x = -2, -1, 0, 1, 2

D: {x | x > 0} R: all real numbers.

Use the given x-values to graph the function. Then graph its inverse. Describe the domain and range of the inverse function. f(x) = 5^x; x = -2, -1, 0, 1, 1.5

about $720.30

Use the graph of the function you wrote to predict the computer's value in 4 years.

about 7.73 billion

Use the graph of the function you wrote to predict the population in 2020.

D: {x > 0}; R: all real numbers

Use the x-values {-2, -1, 0, 1, 2, 3} to graph f(x) = (3/2)^x. Then graph its inverse. Describe the domain and range of the inverse function.

D: {x | x > 0}; R: all real numbers.

Use x = -2, -1, 1, 2 and 3 to graph f(x) = (3/4)^x. Then graph its inverse. Describe the domain and range of the inverse function.

log0.01 = -2

Write the exponential equation in logarithmic form. 10^-2 = 0.01

log^3 27 = 3

Write the exponential equation in logarithmic form. 3^3 = 27

log^3 243 = x

Write the exponential equation in logarithmic form. 3^x = 243

log^4 8 = 1.5

Write the exponential equation in logarithmic form. 4^1.5 = 8

log^9 81 = 2

Write the exponential equation in logarithmic form. 9^2 = 81

log^2.4 1 = 0

Write the exponential function in logarithmic form. 2.4^0 = 1

0.9^2 = 0.81

Write the logarithmic equation in exponential form. log^0.9 0.81 = 2

10^1 = 10

Write the logarithmic equation in exponential form. log^10 10 = 1

12^2 = 144

Write the logarithmic equation in exponential form. log^12 144 = 2

x^3 = -16

Write the logarithmic equation in exponential form. log^x (-16) = 3