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:

Rewrite using substitutions: Given that ln 2 = a, ln 3 = b, and ln 5 = c, we need to express ln((25)/(27)) using these substitutions. First, we can rewrite 25 as 5^2 and 27 as 3^3. So, ln((25)/(27)) becomes ln((5^2)/(3^3)).Separate terms in logarithm: Using the properties of logarithms, we can separate the terms in the logarithm. ln((5^2)/(3^3)) = 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 * ln(x). This gives us 2 * ln(5) - 3 * ln(3).Substitute given values: Substitute the given values for ln(5) and ln(3). This results in 2 * c - 3 * b.Final answer: The expression 2 * c - 3 * b is the final answer in terms of a, b, and c.

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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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Algebra 1

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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 $c$ .