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$Express the given expression as an integer or as a fraction in simplest form. (e^(ln 7-ln 20)) Answer:$

Express the given expression as an integer or as a fraction in simplest form. $\newline$ $\left(e^{\ln 7-\ln 20}\right)$ $\newline$ Answer:

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Csc, sec, and cot of special angles

Full solution

Q. Express the given expression as an integer or as a fraction in simplest form.

\newline

\left(e^{\ln 7-\ln 20}\right)

\newline

Answer:

Rewrite logarithms as division: To simplify the expression $e^{(\ln 7-\ln 20)}$ , we can use the properties of logarithms and exponents. The difference of logarithms $\ln 7 - \ln 20$ can be rewritten as the logarithm of a division: $\ln(\frac{7}{20})$ .
Apply exponential function property: Now we have the expression $e^{\ln(\frac{7}{20})}$ . The exponential function $e^x$ and the natural logarithm $\ln(x)$ are inverse functions. Therefore, $e^{\ln(x)} = x$ for any x > 0.
Simplify expression using inverse property: Applying the inverse property to our expression, we get $e^{\ln(\frac{7}{20})} = \frac{7}{20}$ . This is because $e^{\ln(x)}$ simplifies to $x$ , and in our case, $x$ is $\frac{7}{20}$ .
Final simplified expression: The fraction $\frac{7}{20}$ is already in its simplest form, as $7$ and $20$ have no common factors other than $1$ . Therefore, the expression $(e^{(\ln 7-\ln 20)})$ simplifies to $\frac{7}{20}$ .

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