If In a = 2, In b = 3, and In c =5, evaluate the following:4(a) In63 c2(b) In Vab®c4

Name: If In a = 2, In b = 3, and In c =5, evaluate the following:4(a) In63
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Description: If In a = 2, In b = 3, and In c =5, evaluate the following:4(a) In63 c2(b) In Vab®c4

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Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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If In a = 2, In b = 3, and In c =
5, evaluate the following:
4
(a) In
63 c2
(b) In Vab®c4

Expert Solution

Step 1

GIVEN DATA : The value of $\ln a = 2$ , $\ln b = 3$ and $\ln c = 5$

TO FIND : Evaluate

(a). $\ln (\frac{a^{4}}{b^{3} c^{2}})$

(b). $\ln \sqrt{a^{4} b^{3} c^{4}}$

Step 2

(a). Given expression $\ln (\frac{a^{4}}{b^{3} c^{2}})$

Apply quotient rule of logarithm : $l o g_{m} \frac{a}{b} = l o g_{m} a - l o g_{m} b$

$\ln (\frac{a^{4}}{b^{3} c^{2}})$ $= \ln (a^{4}) - \ln (b^{3} c^{2})$

Apply product rule of logarithm: $l n_{m} (a b) = l n_{m} (a) + l n_{m} (b)$

$\ln (\frac{a^{4}}{b^{3} c^{2}})$ $= \ln (a^{4}) - [\ln (b^{3}) + \ln (c^{2})]$

$\ln (\frac{a^{4}}{b^{3} c^{2}})$ $= \ln (a^{4}) - \ln (b^{3}) - \ln (c^{2})$

Apply power rule of logarithm : $l n_{m} (a^{k}) = k (\ln_{m} (a))$

$\ln (\frac{a^{4}}{b^{3} c^{2}})$ $= 4 \ln (a) - 3 \ln (b) - 2 \ln (c)$

Step 3

Substitute the values of $\ln a = 2$ , $\ln b = 3$ and $\ln c = 5$ from the given data

$\ln (\frac{a^{4}}{b^{3} c^{2}})$ $= 4 (2) - 3 (3) - 2 (5)$

$\ln (\frac{a^{4}}{b^{3} c^{2}})$ $= 8 - 9 - 10$

$l n (\frac{a^{4}}{b^{3} c^{2}})$ $= (- 11)$

Step 4

(b). The given expression is $\ln \sqrt{a^{4} b^{3} c^{4}}$

Apply the exponent rule of power of power : ${(x^{a})}^{b} = x^{a b}$

$\ln \sqrt{a^{4} b^{3} c^{4}}$ $= \ln \sqrt{{(a^{2})}^{2} b^{2} b {(c^{2})}^{2}}$

Square root is expressed as : $x = \sqrt{x^{2}}$

$\ln \sqrt{a^{4} b^{3} c^{4}}$ $= \ln (a^{2} b c^{2} \sqrt{b})$

$\ln \sqrt{a^{4} b^{3} c^{4}}$ $= \ln (a^{2} c^{2} b \sqrt{b})$

$\ln \sqrt{a^{4} b^{3} c^{4}}$ $= \ln (a^{2} c^{2} b {(b)}^{\frac{1}{2}})$

Apply the multiplication rule of exponent : $x^{a} \times x^{b} = x^{a + b}$

$\ln \sqrt{a^{4} b^{3} c^{4}}$ $= \ln (a^{2} c^{2} b^{\frac{3}{2}})$

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