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Given the substitutions
ln 2=a,ln 3=b, and
ln 5=c, find the value of
ln(5sqrt2) 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 $\ln (5 \sqrt{2})$ in terms of $a, b$ , and $c$ . $\newline$ Answer:

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Write equations of cosine functions using properties

Full solution

Q. Given the substitutions

\ln 2=a, \ln 3=b

, and

\ln 5=c

, find the value of

\ln (5 \sqrt{2})

in terms of

a, b

, and

c

.

\newline

Answer:

Break down $\ln(5\sqrt{2})$ : We need to express $\ln(5\sqrt{2})$ using the given substitutions. We can use the properties of logarithms to break down $\ln(5\sqrt{2})$ into parts that include $\ln 2$ and $\ln 5$ .
Use logarithmic properties: Using the property of logarithms that $\ln(xy) = \ln(x) + \ln(y)$ , we can write $\ln(5\sqrt{2})$ as $\ln(5) + \ln(\sqrt{2})$ .
Express $\ln(\sqrt{2})$ : Now, we need to express $\ln(\sqrt{2})$ in terms of $\ln 2$ . Using the property that $\ln(x^y) = y\cdot\ln(x)$ , we can write $\ln(\sqrt{2})$ as $(1/2)\ln(2)$ .
Substitute given values: Substitute the given values for $\ln 2$ and $\ln 5$ into the expression. We have $\ln(5)$ as $c$ and $\ln(2)$ as $a$ , so $\ln(5\sqrt{2})$ becomes $c + (\frac{1}{2})a$ .
Final expression: The final expression for $\ln(5\sqrt{2})$ in terms of $a$ , $b$ , and $c$ is $c + (\frac{1}{2})a$ . There is no need to include $b$ since it does not appear in the expression.

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