Express as a single logarithm. 3log_(a)3+7log_(a)5

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

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

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Algebra 2

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 $\log_{a}(2109375)$ .