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Given the substitutions
ln 2=a,ln 3=b, and
ln 5=c, find the value of
ln((25)/(27)) 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 $\ln \left(\frac{25}{27}\right)$ in terms of $a, b$ , and $c$ . $\newline$ Answer:

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Write and solve inverse variation equations

Full solution

Q. Given the substitutions

\ln 2=a, \ln 3=b

, and

\ln 5=c

, find the value of

\ln \left(\frac{25}{27}\right)

in terms of

a, b

, and

c

.

\newline

Answer:

Rewrite using substitutions: Given that $\ln 2 = a$ , $\ln 3 = b$ , and $\ln 5 = c$ , we need to express $\ln\left(\frac{25}{27}\right)$ using these substitutions. $\newline$ First, we can rewrite $25$ as $5^2$ and $27$ as $3^3$ . $\newline$ So, $\ln\left(\frac{25}{27}\right)$ becomes $\ln\left(\frac{5^2}{3^3}\right)$ .
Separate terms in logarithm: Using the properties of logarithms, we can separate the terms in the logarithm. $\ln\left(\frac{5^2}{3^3}\right) = \ln(5^2) - \ln(3^3)$ .
Apply power rule of logarithms: Now, apply the power rule of logarithms, which states that $\ln(x^y) = y \cdot \ln(x)$ . This gives us $2 \cdot \ln(5) - 3 \cdot \ln(3)$ .
Substitute given values: Substitute the given values for $\ln(5)$ and $\ln(3)$ . This results in $2 \times c - 3 \times b$ .
Final answer: The expression $2 \times c - 3 \times b$ is the final answer in terms of $a$ , $b$ , and $c$ .

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