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Express as a single logarithm. $\newline$ $3\log_{a}3+7\log_{a}5$

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Product property of logarithms

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Q. Express as a single logarithm.

\newline

3\log_{a}3+7\log_{a}5

Identify Properties: Identify the properties of logarithms that can be used to combine the terms. $\newline$ The given expression is $3\log_{a}3 + 7\log_{a}5$ . We can use the power rule of logarithms to move the coefficients in front of the logarithms to the exponent position inside the logarithms. $\newline$ Power Rule: $n\log_{b}(x) = \log_{b}(x^n)$
Apply Power Rule: Apply the power rule to each term in the expression. $\newline$ Using the power rule, we rewrite each term: $\newline$ $3\log_{a}3 = \log_{a}(3^{3})$ $\newline$ $7\log_{a}5 = \log_{a}(5^{7})$
Calculate Exponents: Calculate the exponents. $\newline$ Calculate $3^3$ and $5^7$ : $\newline$ $3^3 = 27$ $\newline$ $5^7 = 78125$ $\newline$ Now the expression is $\log_{a}(27) + \log_{a}(78125)$ .
Combine Logarithms: Combine the logarithms using the product property. $\newline$ Now that we have two logarithms with the same base and no coefficients in front, we can combine them using the product property. $\newline$ Product Property: $\log_b(x) + \log_b(y) = \log_b(xy)$ $\newline$ Combine the logarithms: $\newline$ $\log_{a}(27) + \log_{a}(78125) = \log_{a}(27 \times 78125)$
Calculate Product: Calculate the product inside the logarithm. $\newline$ Multiply $27$ by $78125$ : $\newline$ $27 \times 78125 = 2109375$ $\newline$ Now the expression is $\log_{a}(2109375)$ .

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