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Math  /  Algebra

QuestionProblems 1−61-6, Assuming xx and yy are positive, use properties of logarithms to write the expression as a sum or difference of logarithms.
1. log⁡16x\log 16 x
2. ln⁡(3y)ln⁡3−ln⁡y\ln \left(\frac{3}{y}\right) \ln 3-\ln y
3. ln⁡(e3y5)\ln \left(e^{3} y^{5}\right)
4. log⁡(1000x5)\log \left(1000 x^{5}\right)
5. ln⁡x3y\ln \sqrt{\frac{x^{3}}{y}}
6. log⁡2(x3y2)\log _{2}\left(x^{3} y^{2}\right)
Problems 7 - 12, Assuming x,yx, y, and zz are positive, use properties of logarithms to write the expression as a single logarithm.
7. log⁡264−log⁡24\log _{2} 64-\log _{2} 4
8. ln⁡(x+3)+2ln⁡x\ln (x+3)+2 \ln x
9. 4ln⁡x+7ln⁡y−3ln⁡z4 \ln x+7 \ln y-3 \ln z
10. 13(log⁡x−2log⁡y)\frac{1}{3}(\log x-2 \log y)
11. 13[2log⁡(x+1)−log⁡x−log⁡(x−3)]\frac{1}{3}[2 \log (x+1)-\log x-\log (x-3)]
12. 3[ln⁡x+ln⁡(x−2)]−4ln⁡(x2−4)3[\ln x+\ln (x-2)]-4 \ln \left(x^{2}-4\right)
Problems 13 - 16, Use a calculator to evaluate to three decimal places.
13. log⁡418\log _{4} 18
14. log⁡1223\log _{\frac{1}{2}} 23
15. log⁡π57\log _{\pi} 57
16. log⁡0.816\log _{0.8} 16

Studdy Solution

STEP 1

1. We are using properties of logarithms to manipulate expressions.
2. We assume xx, yy, and zz are positive, allowing us to use logarithmic properties.
3. Calculations for problems 13-16 will be approximated using a calculator.

STEP 2

1. Problems 1-6: Express each logarithmic expression as a sum or difference of logarithms.
2. Problems 7-12: Combine each expression into a single logarithm.
3. Problems 13-16: Evaluate each logarithmic expression to three decimal places using a calculator.

STEP 3

1. Use the product property of logarithms: log⁡16x=log⁡16+log⁡x\log 16x = \log 16 + \log x.
2. Use the quotient property of logarithms: ln⁡(3y)=ln⁡3−ln⁡y\ln \left(\frac{3}{y}\right) = \ln 3 - \ln y.
3. Use the power property of logarithms: ln⁡(e3y5)=ln⁡e3+ln⁡y5=3ln⁡e+5ln⁡y=3+5ln⁡y\ln \left(e^{3} y^{5}\right) = \ln e^{3} + \ln y^{5} = 3\ln e + 5\ln y = 3 + 5\ln y.
4. Use the product property: log⁡(1000x5)=log⁡1000+log⁡x5=3+5log⁡x\log \left(1000 x^{5}\right) = \log 1000 + \log x^{5} = 3 + 5\log x.
5. Use the power and quotient properties: ln⁡x3y=ln⁡(x3y)1/2=12(ln⁡x3−ln⁡y)=32ln⁡x−12ln⁡y\ln \sqrt{\frac{x^{3}}{y}} = \ln \left(\frac{x^{3}}{y}\right)^{1/2} = \frac{1}{2}(\ln x^{3} - \ln y) = \frac{3}{2}\ln x - \frac{1}{2}\ln y.
6. Use the product property: log⁡2(x3y2)=log⁡2x3+log⁡2y2=3log⁡2x+2log⁡2y\log _{2}\left(x^{3} y^{2}\right) = \log _{2}x^{3} + \log _{2}y^{2} = 3\log _{2}x + 2\log _{2}y.

STEP 4

7. Use the quotient property: log⁡264−log⁡24=log⁡2(644)=log⁡216=4\log _{2} 64 - \log _{2} 4 = \log _{2} \left(\frac{64}{4}\right) = \log _{2} 16 = 4.
8. Use the power property: ln⁡(x+3)+2ln⁡x=ln⁡(x+3)+ln⁡x2=ln⁡((x+3)x2)\ln (x+3) + 2 \ln x = \ln (x+3) + \ln x^{2} = \ln \left((x+3)x^{2}\right).
9. Use the power and product properties: 4ln⁡x+7ln⁡y−3ln⁡z=ln⁡x4+ln⁡y7−ln⁡z3=ln⁡(x4y7z3)4 \ln x + 7 \ln y - 3 \ln z = \ln x^{4} + \ln y^{7} - \ln z^{3} = \ln \left(\frac{x^{4}y^{7}}{z^{3}}\right).
10. Use the power and quotient properties: 13(log⁡x−2log⁡y)=13(log⁡x−log⁡y2)=13log⁡(xy2)=log⁡(x1/3y2/3)\frac{1}{3}(\log x - 2 \log y) = \frac{1}{3}(\log x - \log y^{2}) = \frac{1}{3}\log \left(\frac{x}{y^{2}}\right) = \log \left(\frac{x^{1/3}}{y^{2/3}}\right).
11. Use the power and quotient properties: 13[2log⁡(x+1)−log⁡x−log⁡(x−3)]=13log⁡((x+1)2x(x−3))=log⁡((x+1)2/3(x(x−3))1/3)\frac{1}{3}[2 \log (x+1) - \log x - \log (x-3)] = \frac{1}{3}\log \left(\frac{(x+1)^{2}}{x(x-3)}\right) = \log \left(\frac{(x+1)^{2/3}}{(x(x-3))^{1/3}}\right).
12. Use the power and quotient properties: 3[ln⁡x+ln⁡(x−2)]−4ln⁡(x2−4)=ln⁡(x(x−2))3−ln⁡(x2−4)4=ln⁡((x(x−2))3(x2−4)4)3[\ln x + \ln (x-2)] - 4 \ln \left(x^{2}-4\right) = \ln (x(x-2))^{3} - \ln (x^{2}-4)^{4} = \ln \left(\frac{(x(x-2))^{3}}{(x^{2}-4)^{4}}\right).

STEP 5

13. Use a calculator to evaluate: log⁡418≈2.085\log _{4} 18 \approx 2.085.
14. Use a calculator to evaluate: log⁡1223≈−4.523\log _{\frac{1}{2}} 23 \approx -4.523.
15. Use a calculator to evaluate: log⁡π57≈2.364\log _{\pi} 57 \approx 2.364.
16. Use a calculator to evaluate: log⁡0.816≈−4.285\log _{0.8} 16 \approx -4.285.

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