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Given the substitutions
ln 2=a,ln 3=b, and
ln 5=c, find the value of
ln(50) 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 $\ln (50)$ 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 (50)

in terms of

a, b

, and

c

.

\newline

Answer:

Factorize $50$ : We need to express $\ln(50)$ using the given substitutions. We can start by expressing $50$ as a product of its prime factors. $\newline$ $50$ can be written as $2 \times 5^2$ .
Apply Logarithm Property: Now we can use the properties of logarithms to break down $\ln(50)$ into the sum of the logarithms of its prime factors. $\ln(50) = \ln(2 \times 5^2)$
Use Power Rule: Using the logarithm property that $\ln(xy) = \ln(x) + \ln(y)$ , we can separate the factors. $\newline$ $\ln(50) = \ln(2) + \ln(5^2)$
Substitute Given Values: Next, we apply the power rule of logarithms, which states that $\ln(x^y) = y \cdot \ln(x)$ , to the term $\ln(5^2)$ . $\ln(50) = \ln(2) + 2 \cdot \ln(5)$
Substitute Given Values: Next, we apply the power rule of logarithms, which states that $\ln(x^y) = y \cdot \ln(x)$ , to the term $\ln(5^2)$ . $\ln(50) = \ln(2) + 2 \cdot \ln(5)$ Now we substitute the given values for $\ln(2)$ and $\ln(5)$ , which are $a$ and $c$ , respectively. $\ln(50) = a + 2c$

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