Given the substitutions ln 2=a,ln 3=b, and ln 5=c, find the value of ln((root(4)(2))/(3)) in terms of a,b, and c. Answer:

Given Substitutions: We are given the substitutions ln 2 = a, ln 3 = b, and ln 5 = c. We need to express ln((root(4)(2))/(3)) using these substitutions. First, let's rewrite the expression using properties of logarithms and roots. ln((root(4)(2))/(3)) = ln(2^(1/4)) - ln(3)Rewriting Expression: Now, we apply the power rule of logarithms, which states that ln(x^y) = y * ln(x), to the first term. ln(2^(1/4)) = (1/4) * ln(2) Since ln 2 = a, we can substitute a in place of ln(2). (1/4) * ln(2) = (1/4) * aApplying Power Rule: Next, we look at the second term, ln(3). We have been given that ln 3 = b, so we can directly substitute b for ln(3). ln(3) = bSubstituting Values: Now, we combine the two parts of the expression we have simplified: ln((root(4)(2))/(3)) = (1/4) * a - bCombining Simplified Parts: This is the final expression for ln((root(4)(2))/(3)) in terms of a, b, and c. Note that c is not used in this expression because ln(5) does not appear in the original problem.

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Given the substitutions
ln 2=a,ln 3=b, and
ln 5=c, find the value of
ln((root(4)(2))/(3)) in terms of
a,b, and
c.
Answer:

Given the substitutions $\ln 2=a, \ln 3=b$ , and $\ln 5=c$ , find the value of $\ln \left(\frac{\sqrt[4]{2}}{3}\right)$ in terms of $a, b$ , and $c$ . $\newline$ Answer:

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Q. Given the substitutions

\ln 2=a, \ln 3=b

, and

\ln 5=c

, find the value of

\ln \left(\frac{\sqrt[4]{2}}{3}\right)

in terms of

a, b

, and

c

\newline

Answer:

Given Substitutions: We are given the substitutions $\ln 2 = a$ , $\ln 3 = b$ , and $\ln 5 = c$ . We need to express $\ln\left(\frac{\sqrt[4]{2}}{3}\right)$ using these substitutions. $\newline$ First, let's rewrite the expression using properties of logarithms and roots. $\newline$ $\ln\left(\frac{\sqrt[4]{2}}{3}\right) = \ln(2^{1/4}) - \ln(3)$
Rewriting Expression: Now, we apply the power rule of logarithms, which states that $\ln(x^y) = y \cdot \ln(x)$ , to the first term. $\ln(2^{\frac{1}{4}}) = \left(\frac{1}{4}\right) \cdot \ln(2)$ Since $\ln 2 = a$ , we can substitute $a$ in place of $\ln(2)$ . $\left(\frac{1}{4}\right) \cdot \ln(2) = \left(\frac{1}{4}\right) \cdot a$
Applying Power Rule: Next, we look at the second term, $\ln(3)$ . We have been given that $\ln 3 = b$ , so we can directly substitute $b$ for $\ln(3)$ . $\newline$ $\ln(3) = b$
Substituting Values: Now, we combine the two parts of the expression we have simplified: $\newline$ $\ln\left(\frac{\sqrt[4]{2}}{3}\right) = \frac{1}{4} \cdot a - b$
Combining Simplified Parts: This is the final expression for $\ln\left(\frac{\sqrt[4]{2}}{3}\right)$ in terms of $a$ , $b$ , and $c$ . Note that $c$ is not used in this expression because $\ln(5)$ does not appear in the original problem.