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Given the substitutions
ln 2=a,ln 3=b, and
ln 5=c, find the value of
ln(200) 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 $\ln (200)$ 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 (200)

in terms of

a, b

, and

c

.

\newline

Answer:

Factor and Expand: Express $\ln(200)$ in terms of $\ln(2)$ , $\ln(3)$ , and $\ln(5)$ . We can factor $200$ as $2^3 \times 5^2$ . Therefore, $\ln(200)$ can be written as $\ln(2^3 \times 5^2)$ . Using the properties of logarithms, we can expand this as $\ln(2^3) + \ln(5^2)$ .
Apply Power Rule: Apply the power rule of logarithms to simplify $\ln(2^3)$ and $\ln(5^2)$ . The power rule states that $\ln(x^n) = n \cdot \ln(x)$ . Therefore, $\ln(2^3)$ becomes $3 \cdot \ln(2)$ and $\ln(5^2)$ becomes $2 \cdot \ln(5)$ .
Substitute Values: Substitute the given values for $\ln(2)$ and $\ln(5)$ . We know that $\ln(2) = a$ and $\ln(5) = c$ . So, we replace $\ln(2)$ with $a$ and $\ln(5)$ with $c$ in our equation. This gives us $3 \times a + 2 \times c$ .
Final Expression: Write the final expression for $\ln(200)$ in terms of $a$ , $b$ , and $c$ . $\newline$ Since there is no $\ln(3)$ in our expression, $b$ does not appear in the final answer. $\newline$ The final expression for $\ln(200)$ is $3 \times a + 2 \times c$ .

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