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Write the expression
3ln 3-2ln 5 as a single logarithm in simplest form without any negative exponents.
Answer:
ln(◻)

Write the expression $3 \ln 3-2 \ln 5$ as a single logarithm in simplest form without any negative exponents. $\newline$ Answer: $\ln (\square)$

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Change of base formula

Full solution

Q. Write the expression

3 \ln 3-2 \ln 5

as a single logarithm in simplest form without any negative exponents.

\newline

Answer:

\ln (\square)

Apply Power Rule: Apply the power rule of logarithms to rewrite the expression. $\newline$ The power rule states that $a \cdot \log(b) = \log(b^a)$ . We can apply this to both terms in the expression. $\newline$ $3\ln(3)$ becomes $\ln(3^3)$ and $2\ln(5)$ becomes $\ln(5^2)$ .
Rewrite Using Power Rule: Rewrite the expression using the power rule. $\ln(3^3) - \ln(5^2)$ becomes $\ln(27) - \ln(25)$ .
Apply Quotient Rule: Apply the quotient rule of logarithms to combine the two logarithms into one. $\newline$ The quotient rule states that $\log(b) - \log(c) = \log\left(\frac{b}{c}\right)$ . We can apply this to combine $\ln(27)$ and $\ln(25)$ . $\newline$ $\ln(27) - \ln(25)$ becomes $\ln\left(\frac{27}{25}\right)$ .
Simplify Fraction: Simplify the fraction inside the logarithm if possible. $\newline$ The fraction $\frac{27}{25}$ is already in simplest form, so no further simplification is needed.

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