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Given the substitutions
ln 2=a,ln 3=b, and
ln 5=c, find the value of
ln(4sqrt5) 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 $\ln (4 \sqrt{5})$ 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 (4 \sqrt{5})

in terms of

a, b

, and

c

.

\newline

Answer:

Breakdown using properties: We need to express $\ln(4\sqrt{5})$ in terms of $a$ , $b$ , and $c$ . We can start by using the properties of logarithms to break down $\ln(4\sqrt{5})$ into simpler parts that involve $\ln(2)$ and $\ln(5)$ . $\newline$ $\ln(4\sqrt{5}) = \ln(4) + \ln(\sqrt{5})$
Express $\ln(4)$ : We know that $4$ is $2$ squared, so $\ln(4)$ can be written as $\ln(2^2)$ . Using the power rule for logarithms, which states that $\ln(x^y) = y\cdot\ln(x)$ , we get: $\newline$ $\ln(4) = \ln(2^2) = 2\cdot\ln(2)$ $\newline$ Since $\ln(2) = a$ , we can substitute $a$ for $\ln(2)$ : $\newline$ $4$ $0$
Express $\ln(\sqrt{5})$ : Next, we look at $\ln(\sqrt{5})$ . The square root of $5$ is the same as $5$ raised to the $\frac{1}{2}$ power. Using the power rule for logarithms again, we get: $\newline$ $\ln(\sqrt{5}) = \ln(5^{\frac{1}{2}}) = (\frac{1}{2})\cdot\ln(5)$ $\newline$ Since $\ln(5) = c$ , we can substitute $c$ for $\ln(5)$ : $\newline$ $\ln(\sqrt{5}) = (\frac{1}{2})\cdot c$
Combine results: Now we can combine our results for $\ln(4)$ and $\ln(\sqrt{5})$ to find $\ln(4\sqrt{5})$ : $\ln(4\sqrt{5}) = \ln(4) + \ln(\sqrt{5}) = 2\cdot a + \left(\frac{1}{2}\right)\cdot c$
Final expression: We have successfully expressed $\ln(4\sqrt{5})$ in terms of $a$ and $c$ . There is no need to include $b$ in our expression because it does not appear in the original problem.

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