3.7.1 Classwork

Solve.

1. [latex]\log_7(x-2)=1[/latex]
2. [latex]\log_2(2x+6)=1[/latex]
3. [latex]\log_6(x-2)=3[/latex]
4. [latex]2\log_7(x-2)=1[/latex]
5. [latex]\log_2(2x+6)+2=1[/latex]
6. [latex]\log_6(x-2)-2=0[/latex]

Answer the following.

7. How do you solve [latex]y=e^x[/latex] for x?
8. How do you solve [latex]y=\log(x)[/latex] for x?
9. How do you solve [latex]y=\log_3(x)[/latex] for x?

The Power Property for Logarithms: [latex]\log_b(x^p)=p\log_b(x)[/latex], where [latex]b>0[/latex]

Use the Power Property for Logarithms.

10. [latex]\ln(x^3)[/latex]
11. [latex]\log\left[{(x-2)}^5\right][/latex]
12. [latex]\log_5\left(\sqrt x\right)[/latex]
13. [latex]In\;\left(\sqrt[5]{x^2}\right)[/latex]
14. [latex]\ln\left(b^c\right)[/latex]

Solve. Round approximate answers to three decimal places.

15. [latex]250=500{(1-0.07)}^t[/latex]
16. [latex]100=500\left(1-\frac{0.07}{12}\right)^{12t}[/latex]
17. [latex]1,000=500{(1+0.01)}^t[/latex]
18. [latex]\log_2(4x-1)=-3[/latex]
19. [latex]\log_3(2x+1)=4[/latex]
20. [latex]5\log_4(5-2x)=-10[/latex]

Solve.

If [latex]a^u=a^v[/latex], then [latex]u=v[/latex]. Convert each side of the equation to the same base, then solve.

21. [latex]2^{x-4}=2^{3x+7}[/latex]
22. [latex]3^{2x}=27^{x-1}[/latex]
23. [latex]2^{2x-1}=32[/latex]
24. [latex]5^x=625[/latex]
25. [latex]4^{2x-1}=64[/latex]
26. [latex]32^x=8[/latex]
27. [latex]4^x=\frac1{16}[/latex]
28. [latex]8=\frac1{4^x}[/latex]

3.7.2 Homework

Solve for the exact answer.

1. [latex]10^x=34,926[/latex]
2. [latex]e^x=64[/latex]
3. [latex]3^7=x[/latex]
4. [latex]4^x=1024[/latex]
5. [latex]3\ln(5x)=9[/latex]
6. [latex]e^{4x-5}-7=11,243[/latex]
7. [latex]e^{1-8x}=7957[/latex]
8. [latex]4e^{7x}=10,273[/latex]
9. [latex]9^x=27[/latex]
10. [latex]5^{2-x}=\frac1{125}[/latex]
11. [latex]3^{1-x}=\frac1{27}[/latex]
12. [latex]250=50e^{0.06\cdot t}[/latex]
13. [latex]25=50e^{k7}[/latex]
14. [latex]25=50e^{k10000}[/latex]
15. [latex]1000=500{(1+.01)}^t[/latex]
16. [latex]\log(4x)=2[/latex]
17. [latex]5{(2)}^{3x}=20[/latex]
18. [latex]4^{3x}=6[/latex]
19. [latex]\log(2x+5)=2[/latex]
20. [latex]\log(5x)=3[/latex]

Solve. Round approximate answers to two decimal places.

21. If your money grew to [latex]\$12,000[/latex] invested at simple interest on an investment of [latex]\$10,000[/latex] in five years, what is the rate of interest?
22. How long would it take your brand new car to depreciate [latex]40\%?[/latex] Assume that it depreciates at [latex]15\%[/latex] per year (compounded continuously).
    (If you lost [latex]40\%[/latex], what would A=?)
23. The population of a town increases by [latex]P(t)=2500e^{0.0293t}[/latex] where t is time in years since 1990. Find the population in each of the following years.
    1. [latex]2000[/latex]
    2. [latex]2010[/latex]
    3. [latex]2018[/latex]
24. The population of a city in 2000 was 240,360, and the exponential growth rate was [latex]1.2\%[/latex]. Write the exponential growth function. When will the population be 275,000?
25. Bacteria growth is modeled by this function where N is the number of bacteria in thousands, and t is the time in minutes: [latex]N=e^{kt}[/latex] If the number of bacteria is 570,000 when time is 4 minutes, find the value for k, the rate of growth.
26. Radioactive decay is modeled by the function where R is the ratio of remaining material, k is the rate of decay, and t is time in years: [latex]R=e^{kt}[/latex]
    1. Find k at 800 years when R=0.80 ([latex]80\%[/latex] of its radioactive energy).
    2. Find the half-life of this element. (Half-life means the material still has [latex]50\%[/latex] of its life left).
27. At the end of five years, an initial investment of [latex]\$7,000[/latex] is now worth [latex]\$8,549.82[/latex]. Assuming that interest was compounded continuously, what was the rate?
28. At the end of seven years, an initial investment of [latex]\$15,000[/latex] is now worth [latex]\$19,502.05[/latex]. Assuming that interest was compounded continuously, what was the rate?
29. What would you have to invest in order to reach [latex]\$10,000[/latex] in 5 years at a rate of [latex]2.5\%[/latex], compounded continuously?
30. If Free Yer Assets Bank (FYAB) will give you [latex]0.25\%[/latex] compounded quarterly and [latex]\text{I}.\text{M}.\text{A}.\text{Q}.\text{T}.\mathrm\pi[/latex] bank will give you [latex]0.23\%[/latex] interest compounded continuously, which is offering the better deal (Hint: Show how long it takes to double an investment).
31. After exercising, your heart rate can be modeled by [latex]R\left(t\right)=151e^{-0.055t}[/latex] where [latex]0\leq t\leq15[/latex] and t is the number of minutes that have elapsed after you stop exercising.
    1. How many minutes does it take your heart rate to drop to 100 beats per minute?
    2. Find your heart rate 15 minutes after exercising.