Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Experience the convenience of finding accurate answers to your questions from knowledgeable professionals on our platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
Let's evaluate each expression step-by-step.
### Given:
[tex]\[ \ln a = 2, \quad \ln b = 3, \quad \ln c = 5 \][/tex]
### (a) Evaluate [tex]\(\ln \left(\frac{a^2}{b^2 c^3}\right)\)[/tex]
First, use the properties of logarithms to express the given term:
[tex]\[ \ln \left( \frac{a^2}{b^2 c^3} \right) = \ln(a^2) - \ln(b^2) - \ln(c^3) \][/tex]
Now, calculate each term individually:
[tex]\[ \ln(a^2) = 2 \cdot \ln(a) = 2 \cdot 2 = 4 \][/tex]
[tex]\[ \ln(b^2) = 2 \cdot \ln(b) = 2 \cdot 3 = 6 \][/tex]
[tex]\[ \ln(c^3) = 3 \cdot \ln(c) = 3 \cdot 5 = 15 \][/tex]
Combine these results:
[tex]\[ \ln \left( \frac{a^2}{b^2 c^3} \right) = 4 - 6 - 15 = -17 \][/tex]
Thus, the answer is:
[tex]\[ \boxed{-17} \][/tex]
### (b) Evaluate [tex]\(\ln \sqrt{b^{-3} c a^{-1}}\)[/tex]
First, use the properties of logarithms to express the given term:
[tex]\[ \ln \sqrt{b^{-3} c a^{-1}} = \frac{1}{2} \ln(b^{-3} c a^{-1}) \][/tex]
Now, use logarithm properties again:
[tex]\[ \ln(b^{-3} c a^{-1}) = \ln(b^{-3}) + \ln(c) + \ln(a^{-1}) \][/tex]
[tex]\[ \ln(b^{-3}) = -3 \cdot \ln(b) = -3 \cdot 3 = -9 \][/tex]
[tex]\[ \ln(a^{-1}) = -1 \cdot \ln(a) = -1 \cdot 2 = -2 \][/tex]
Combine these results:
[tex]\[ \ln(b^{-3} c a^{-1}) = -9 + 5 - 2 = -6 \][/tex]
Then, evaluate:
[tex]\[ \ln \sqrt{b^{-3} c a^{-1}} = \frac{1}{2} \cdot (-6) = -3 \][/tex]
Thus, the answer is:
[tex]\[ \boxed{-3} \][/tex]
### (c) Evaluate [tex]\(\frac{\ln \left(a^{-1} b^{-3}\right)}{\ln (b c)^{-2}}\)[/tex]
First, use the properties of logarithms to express the given term:
[tex]\[ \ln(a^{-1} b^{-3}) = \ln(a^{-1}) + \ln(b^{-3}) \][/tex]
[tex]\[ \ln(a^{-1}) = -\ln(a) = -2 \][/tex]
[tex]\[ \ln(b^{-3}) = -3 \cdot \ln(b) = -3 \cdot 3 = -9 \][/tex]
Combine these results:
[tex]\[ \ln(a^{-1} b^{-3}) = -2 - 9 = -11 \][/tex]
Next, consider the denominator:
[tex]\[ \ln(b c)^{-2} = -2 \cdot \ln(b c) \][/tex]
[tex]\[ \ln(b c) = \ln(b) + \ln(c) = 3 + 5 = 8 \][/tex]
[tex]\[ \ln(b c)^{-2} = -2 \cdot 8 = -16 \][/tex]
Combine the results to find the original expression:
[tex]\[ \frac{\ln(a^{-1} b^{-3})}{\ln(b c)^{-2}} = \frac{-11}{-16} = \frac{11}{16} \approx 0.6875 \][/tex]
Thus, the answer is:
[tex]\[ \boxed{0.6875} \][/tex]
### (d) Evaluate [tex]\(\left(\ln c^4\right) \left( \ln \frac{a}{b^{-4}} \right)^{-2}\)[/tex]
First, use the properties of logarithms to express the given term:
[tex]\[ \ln(c^4) = 4 \cdot \ln(c) = 4 \cdot 5 = 20 \][/tex]
Next, consider:
[tex]\[ \ln \left(\frac{a}{b^{-4}}\right) = \ln(a) - \ln(b^{-4}) \][/tex]
[tex]\[ \ln(b^{-4}) = -4 \cdot \ln(b) = -4 \cdot 3 = -12 \][/tex]
Combine these results:
[tex]\[ \ln \frac{a}{b^{-4}} = \ln(a) + 12 = 2 + 12 = 14 \][/tex]
Finally, evaluate the given term:
[tex]\[ \left( \ln c^4 \right) \left( \ln \frac{a}{b^{-4}} \right)^{-2} = 20 \cdot \left( 14 \right)^{-2} = 20 \cdot \left( \frac{1}{196} \right) = \frac{20}{196} = \frac{10}{98} = \frac{5}{49} \approx 0.1020408163265306 \][/tex]
Thus, the answer is:
[tex]\[ \boxed{0.1020408163265306} \][/tex]
These evaluations confirm the results step-by-step for each given expression.
### Given:
[tex]\[ \ln a = 2, \quad \ln b = 3, \quad \ln c = 5 \][/tex]
### (a) Evaluate [tex]\(\ln \left(\frac{a^2}{b^2 c^3}\right)\)[/tex]
First, use the properties of logarithms to express the given term:
[tex]\[ \ln \left( \frac{a^2}{b^2 c^3} \right) = \ln(a^2) - \ln(b^2) - \ln(c^3) \][/tex]
Now, calculate each term individually:
[tex]\[ \ln(a^2) = 2 \cdot \ln(a) = 2 \cdot 2 = 4 \][/tex]
[tex]\[ \ln(b^2) = 2 \cdot \ln(b) = 2 \cdot 3 = 6 \][/tex]
[tex]\[ \ln(c^3) = 3 \cdot \ln(c) = 3 \cdot 5 = 15 \][/tex]
Combine these results:
[tex]\[ \ln \left( \frac{a^2}{b^2 c^3} \right) = 4 - 6 - 15 = -17 \][/tex]
Thus, the answer is:
[tex]\[ \boxed{-17} \][/tex]
### (b) Evaluate [tex]\(\ln \sqrt{b^{-3} c a^{-1}}\)[/tex]
First, use the properties of logarithms to express the given term:
[tex]\[ \ln \sqrt{b^{-3} c a^{-1}} = \frac{1}{2} \ln(b^{-3} c a^{-1}) \][/tex]
Now, use logarithm properties again:
[tex]\[ \ln(b^{-3} c a^{-1}) = \ln(b^{-3}) + \ln(c) + \ln(a^{-1}) \][/tex]
[tex]\[ \ln(b^{-3}) = -3 \cdot \ln(b) = -3 \cdot 3 = -9 \][/tex]
[tex]\[ \ln(a^{-1}) = -1 \cdot \ln(a) = -1 \cdot 2 = -2 \][/tex]
Combine these results:
[tex]\[ \ln(b^{-3} c a^{-1}) = -9 + 5 - 2 = -6 \][/tex]
Then, evaluate:
[tex]\[ \ln \sqrt{b^{-3} c a^{-1}} = \frac{1}{2} \cdot (-6) = -3 \][/tex]
Thus, the answer is:
[tex]\[ \boxed{-3} \][/tex]
### (c) Evaluate [tex]\(\frac{\ln \left(a^{-1} b^{-3}\right)}{\ln (b c)^{-2}}\)[/tex]
First, use the properties of logarithms to express the given term:
[tex]\[ \ln(a^{-1} b^{-3}) = \ln(a^{-1}) + \ln(b^{-3}) \][/tex]
[tex]\[ \ln(a^{-1}) = -\ln(a) = -2 \][/tex]
[tex]\[ \ln(b^{-3}) = -3 \cdot \ln(b) = -3 \cdot 3 = -9 \][/tex]
Combine these results:
[tex]\[ \ln(a^{-1} b^{-3}) = -2 - 9 = -11 \][/tex]
Next, consider the denominator:
[tex]\[ \ln(b c)^{-2} = -2 \cdot \ln(b c) \][/tex]
[tex]\[ \ln(b c) = \ln(b) + \ln(c) = 3 + 5 = 8 \][/tex]
[tex]\[ \ln(b c)^{-2} = -2 \cdot 8 = -16 \][/tex]
Combine the results to find the original expression:
[tex]\[ \frac{\ln(a^{-1} b^{-3})}{\ln(b c)^{-2}} = \frac{-11}{-16} = \frac{11}{16} \approx 0.6875 \][/tex]
Thus, the answer is:
[tex]\[ \boxed{0.6875} \][/tex]
### (d) Evaluate [tex]\(\left(\ln c^4\right) \left( \ln \frac{a}{b^{-4}} \right)^{-2}\)[/tex]
First, use the properties of logarithms to express the given term:
[tex]\[ \ln(c^4) = 4 \cdot \ln(c) = 4 \cdot 5 = 20 \][/tex]
Next, consider:
[tex]\[ \ln \left(\frac{a}{b^{-4}}\right) = \ln(a) - \ln(b^{-4}) \][/tex]
[tex]\[ \ln(b^{-4}) = -4 \cdot \ln(b) = -4 \cdot 3 = -12 \][/tex]
Combine these results:
[tex]\[ \ln \frac{a}{b^{-4}} = \ln(a) + 12 = 2 + 12 = 14 \][/tex]
Finally, evaluate the given term:
[tex]\[ \left( \ln c^4 \right) \left( \ln \frac{a}{b^{-4}} \right)^{-2} = 20 \cdot \left( 14 \right)^{-2} = 20 \cdot \left( \frac{1}{196} \right) = \frac{20}{196} = \frac{10}{98} = \frac{5}{49} \approx 0.1020408163265306 \][/tex]
Thus, the answer is:
[tex]\[ \boxed{0.1020408163265306} \][/tex]
These evaluations confirm the results step-by-step for each given expression.
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
