MATH 110

Advanced Functions

1) Solve: 4096x = 8 1)

2) Solve: 1 9

x = 729 2)

3) Solve: 4 7

x = 2401

256 3)

4) Solve: 4-x = 1 256

4)

5) Solve: 4(8 - 2x) = 256 5)

6) Solve: 3(6 - 3x) = 1 27

6)

7) Solve: 3(6 + 3x) = 1 27

7)

8) Solve: 4 = b2/3 8)

9) Solve: a3/4 = 125 9)

10) Solve: 25 9

x+1 = 3

5 x-1

10)

11) Solve: e4x - 1 = (e3)-x 11)

12) Solve: 8x - 1 = 323x 12)

13) Solve: m-4 = 1 81

13)

14) Solve: 1 3

2x + 3 = 9x- 5 14)

1

15) Solve: ( 5 )x + 1 = 25x 15)

16) Solve: ex - 3 = 1 e6

x + 2 16)

17) The growth in the mouse population at a certain county dump can be modeled by the exponential function A(t)= 906e0.012t, where t is the number of months since the population was first recorded. Estimate the population after 36 months.

17)

18) The decay of 938 mg of an isotope is given by A(t)= 938e-0.022t, where t is time in years since the initial amount of 938 mg was present. Find the amount (to the nearest milligram) left after 96 years.

18)

19) The sales of a mature product (one which has passed its peak) will decline according to the function S(t) = S0e-at, where t is time in years since the peak sales. Find the sales of a product 17 years after its peak sales if a = 0.22 and S0 = 77,500.

19)

20) The number of reports of a certain virus has increased exponentially since year 0. The number of cases can be approximated using the function r(t) = 119 e0.008t, where t is the number of years since year 0. Estimate the number of cases in year 40.

20)

21) An element decays at the rate of S(t) = se-0.048t, where s is the initial amount in grams and t is the time in years since this initial amount was present. If you have a 71-gram piece of this element, how many grams will you have 5 years from now? Round your answer to the nearest tenth of a gram.

21)

22) Solve: log5 125 = x 22)

23) Solve: log3 1 27

= x 23)

24) Solve: log7 712 = x 24)

25) Solve: logx 625 = 4 25)

2

26) Solve: x = 8log8 13 26)

27) Solve: x = log10 0.01 27)

28) Solve: x = log2 5

8 28)

29) Solve: logx 9 = - 2 29)

30) Solve: log4 x = 3 30)

31) Solve: log5 x = -3 31)

32) Solve: log(x - 5) 10 = 1 32)

33) Solve: log(x + 7) 11 = 1 33)

34) Solve: 8x - 32 = logx 1 34)

35) Suppose f(x) = logax and f(4) = 2. Find f(16). 35)

36) Suppose f(x) = logax and f(4) = 2. Find f 1 16

. 36)

37) Suppose f(x) = loga(x) and f(7) = 2. Find f(343). 37)

38) Suppose f(x) = loga(x) and f(5) = 2. Find f 5 5

38)

39) Evaluate: 100log107 39)

40) Evaluate: log10(0.01)9 40)

41) Evaluate: 1000log1010 41)

42) Evaluate: log10(0.0001)10 42)

3

43) The growth in population of a city can be seen using the formula p(t) = 2148e0.008t, where t is the number of years. Use this formula to calculate the population after 10 years.

43)

44) Suppose the government wants to impose a tax on fossil fuels to reduce carbon emissions. The cost benefit is modeled by ln(1 - P) = -0.0039 - 0.0051x, where x represents the dollars of tax per ton of carbon emitted and P represents the percent reduction in emissions of carbon. (P is in decimal form.) Determine P when x = 63. Round to three decimal places.

44)

45) Suppose f(x) = 32.6 + 1.2log (x + 1) models salinity of ocean water to depths of 1000 meters at a certain latitude. x is the depth in meters and f(x) is in grams of salt per kilogram of seawater. Approximate the salinity (to the nearest hundredth) when the depth is 771 meters.

45)

4

Answer Key Testname: FORUM ADVANCED FUNCTIONS

1) 1 4

2) {-3} 3) {-4} 4) {4} 5) {2} 6) {3} 7) {-3} 8) {-8, 8} 9) {625}

10) - 1 3

11) 1 7

12) - 1 4

13) {-3, 3}

14) 7 4

15) 1 3

16) - 9 7

17) 1396 18) 113 19) 1841 units 20) 164 cases 21) 55.9 g 22) {3} 23) {-3} 24) {6} 25) {5} 26) {13} 27) {-2}

28) 3 5

29) 1 3

30) {64}

5

Answer Key Testname: FORUM ADVANCED FUNCTIONS

31) 1 125

32) {15} 33) {4} 34) {4} 35) 4 36) -4 37) 6 38) -1 39) 49 40) -18 41) 1000 42) -40 43) 2327 44) 0.278 45) 36.07 g/kg

6

Using the Equation Editor …

… to make math problems look like math problems!

Math on the keyboard:

x = [-b +/- sqrt(b^2 – 4ac)]/(2a)

Huh?

Math with the equation editor!

2 4

2

b b ac x

a

−  − =

That’s better!

Let’s say that you want to show an algebraic fraction like

Trying to do this with just the keyboard looks “weird” …

x / (x + 3)

… and it invites mistakes (like not using the parentheses).

x / x + 3 means 𝑥

𝑥 + 3 which is something completely different!

3

x

x +

The next time you go to post a discussion click the three extra dots at the end.

You will see more options show up.

Right below the Bold button you will see the Greek letter sigma.

This is the button to open the equation editor.

Select Graphical Equation

The Equation Editor has lots of tools, so let’s look at a few of the basics.

Good news: most things look like what they do.

For example, if you need a fraction, you will click the button that looks like an empty fraction.

You will see a fraction appear - you just need to fill it in.

Use the arrows to navigate to the bottom of the fraction.

Be sure to use the right arrow to navigate out of the fraction before continuing with your equation.

You will notice that the cursor is tall in this picture. That is how you know that you are out of the fraction.

Another commonly used symbol is the exponent.

This time let’s try to create the expression:

To do this you will start by creating a fraction.

2

6

5

2

x

y

−

+

Once you are in the fraction, you type the x and then click the exponent

button and then the 2.

Use the right arrow button to get out of the exponent before typing the – 5.

Then, use the down arrow to move to the denominator. In this picture you can see the cursor. Note that the cursor will show you if

you are still in the exponent or not, so watch carefully!

Once you have the top of the fraction done …

… just rinse and repeat for the bottom!

Once done click insert.

We did it!

If you want to add words to your equations, just type them in,

select them, and then click on the text font.

Experiment with it until

you can get the desired results.

The math may still be a challenge …

… but now writing the answers shouldn’t be!

Advanced Functions

1) Solve: 4096x = 8 1)

2) Solve: 1 9

x = 729 2)

3) Solve: 4 7

x = 2401

256 3)

4) Solve: 4-x = 1 256

4)

5) Solve: 4(8 - 2x) = 256 5)

6) Solve: 3(6 - 3x) = 1 27

6)

7) Solve: 3(6 + 3x) = 1 27

7)

8) Solve: 4 = b2/3 8)

9) Solve: a3/4 = 125 9)

10) Solve: 25 9

x+1 = 3

5 x-1

10)

11) Solve: e4x - 1 = (e3)-x 11)

12) Solve: 8x - 1 = 323x 12)

13) Solve: m-4 = 1 81

13)

14) Solve: 1 3

2x + 3 = 9x- 5 14)

1

15) Solve: ( 5 )x + 1 = 25x 15)

16) Solve: ex - 3 = 1 e6

x + 2 16)

17) The growth in the mouse population at a certain county dump can be modeled by the exponential function A(t)= 906e0.012t, where t is the number of months since the population was first recorded. Estimate the population after 36 months.

17)

18) The decay of 938 mg of an isotope is given by A(t)= 938e-0.022t, where t is time in years since the initial amount of 938 mg was present. Find the amount (to the nearest milligram) left after 96 years.

18)

19) The sales of a mature product (one which has passed its peak) will decline according to the function S(t) = S0e-at, where t is time in years since the peak sales. Find the sales of a product 17 years after its peak sales if a = 0.22 and S0 = 77,500.

19)

20) The number of reports of a certain virus has increased exponentially since year 0. The number of cases can be approximated using the function r(t) = 119 e0.008t, where t is the number of years since year 0. Estimate the number of cases in year 40.

20)

21) An element decays at the rate of S(t) = se-0.048t, where s is the initial amount in grams and t is the time in years since this initial amount was present. If you have a 71-gram piece of this element, how many grams will you have 5 years from now? Round your answer to the nearest tenth of a gram.

21)

22) Solve: log5 125 = x 22)

23) Solve: log3 1 27

= x 23)

24) Solve: log7 712 = x 24)

25) Solve: logx 625 = 4 25)

2

26) Solve: x = 8log8 13 26)

27) Solve: x = log10 0.01 27)

28) Solve: x = log2 5

8 28)

29) Solve: logx 9 = - 2 29)

30) Solve: log4 x = 3 30)

31) Solve: log5 x = -3 31)

32) Solve: log(x - 5) 10 = 1 32)

33) Solve: log(x + 7) 11 = 1 33)

34) Solve: 8x - 32 = logx 1 34)

35) Suppose f(x) = logax and f(4) = 2. Find f(16). 35)

36) Suppose f(x) = logax and f(4) = 2. Find f 1 16

. 36)

37) Suppose f(x) = loga(x) and f(7) = 2. Find f(343). 37)

38) Suppose f(x) = loga(x) and f(5) = 2. Find f 5 5

38)

39) Evaluate: 100log107 39)

40) Evaluate: log10(0.01)9 40)

41) Evaluate: 1000log1010 41)

42) Evaluate: log10(0.0001)10 42)

3

43) The growth in population of a city can be seen using the formula p(t) = 2148e0.008t, where t is the number of years. Use this formula to calculate the population after 10 years.

43)

44) Suppose the government wants to impose a tax on fossil fuels to reduce carbon emissions. The cost benefit is modeled by ln(1 - P) = -0.0039 - 0.0051x, where x represents the dollars of tax per ton of carbon emitted and P represents the percent reduction in emissions of carbon. (P is in decimal form.) Determine P when x = 63. Round to three decimal places.

44)

45) Suppose f(x) = 32.6 + 1.2log (x + 1) models salinity of ocean water to depths of 1000 meters at a certain latitude. x is the depth in meters and f(x) is in grams of salt per kilogram of seawater. Approximate the salinity (to the nearest hundredth) when the depth is 771 meters.

45)

4

Answer Key Testname: FORUM ADVANCED FUNCTIONS

1) 1 4

2) {-3} 3) {-4} 4) {4} 5) {2} 6) {3} 7) {-3} 8) {-8, 8} 9) {625}

10) - 1 3

11) 1 7

12) - 1 4

13) {-3, 3}

14) 7 4

15) 1 3

16) - 9 7

17) 1396 18) 113 19) 1841 units 20) 164 cases 21) 55.9 g 22) {3} 23) {-3} 24) {6} 25) {5} 26) {13} 27) {-2}

28) 3 5

29) 1 3

30) {64}

5

Answer Key Testname: FORUM ADVANCED FUNCTIONS

31) 1 125

32) {15} 33) {4} 34) {4} 35) 4 36) -4 37) 6 38) -1 39) 49 40) -18 41) 1000 42) -40 43) 2327 44) 0.278 45) 36.07 g/kg

6