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

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Write and solve direct variation equations

Full solution

Q. Given the substitutions

\ln 2=a, \ln 3=b

, and

\ln 5=c

, find the value of

\ln \left(\frac{e}{3}\right)

in terms of

a, b

, and

c

.

\newline

Answer:

Express $\ln\left(\frac{e}{3}\right)$ : Express $\ln\left(\frac{e}{3}\right)$ using the properties of logarithms. $\newline$ We know that $\ln(e) = 1$ because the natural logarithm of $e$ to the power of $1$ is $e$ . We also know that $\ln(3) = b$ . Using the quotient rule for logarithms, which states that $\ln\left(\frac{x}{y}\right) = \ln(x) - \ln(y)$ , we can write $\ln\left(\frac{e}{3}\right)$ as $\ln(e) - \ln(3)$ .
Substitute known values: Substitute the known values into the expression. $\newline$ Substitute $\ln(e) = 1$ and $\ln(3) = b$ into the expression from Step $1$ . $\newline$ $\ln\left(\frac{e}{3}\right) = \ln(e) - \ln(3) = 1 - b$
Simplify expression: Simplify the expression. $\newline$ Simplify the expression to get the final answer in terms of $a$ , $b$ , and $c$ . $\newline$ $\ln\left(\frac{e}{3}\right) = 1 - b$ $\newline$ Since there are no terms involving $a$ or $c$ , the expression remains as is.

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