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Write the expression
(1)/(2)ln 9+ln 3 as a single logarithm in simplest form without any negative exponents.
Answer:
ln(◻)

Write the expression $\frac{1}{2} \ln 9+\ln 3$ as a single logarithm in simplest form without any negative exponents. $\newline$ Answer: $\ln (\square)$

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Power rule with rational exponents

Full solution

Q. Write the expression

\frac{1}{2} \ln 9+\ln 3

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

\newline

Answer:

\ln (\square)

Apply Power Rule: We have the expression $(\frac{1}{2})\ln 9 + \ln 3$ . To combine these logarithms into a single logarithm, we can use the logarithm properties.
Calculate Square Root: First, we apply the power rule of logarithms to the term $(\frac{1}{2})\ln 9$ , which states that a coefficient in front of a logarithm can be turned into an exponent inside the logarithm: $a \cdot \ln(b) = \ln(b^a)$ . $\newline$ $(\frac{1}{2})\ln 9 = \ln(9^{(\frac{1}{2})})$
Combine Logarithms: We know that $9^{\frac{1}{2}}$ is the square root of $9$ , which is $3$ . $\newline$ $\ln(9^{\frac{1}{2}}) = \ln(3)$
Calculate Product: Now we have $\ln(3) + \ln(3)$ . We can combine these using the product rule for logarithms, which states that $\ln(a) + \ln(b) = \ln(a \times b)$ . $\newline$ $\ln(3) + \ln(3) = \ln(3 \times 3)$
Final Simplification: We calculate the product inside the logarithm. $\newline$ $3 \times 3 = 9$ $\newline$ $\ln(3 \times 3) = \ln(9)$
Final Simplification: We calculate the product inside the logarithm. $\newline$ $3 \times 3 = 9$ $\newline$ $\ln(3 \times 3) = \ln(9)$ The expression is now simplified to a single logarithm. $\newline$ $\ln(9)$

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