Identify Relationship: Identify the relationship between the base of the logarithm and the number. The base of the logarithm is 9, and we need to find the exponent that 9 must be raised to in order to get 729.Express as Power: Express 729 as a power of 9. We know that 9 is 3 squared (9 = 3^2), and 729 is 3 to the sixth power (729 = 3^6). Therefore, 729 can be expressed as (3^2)^3.Rewrite Logarithm: Rewrite the logarithm using the new expression for 729. log_(9)(729) becomes log_(9)((3^2)^3).Apply Power Property: Apply the power property of logarithms. The power property states that log_b(P^Q) = Q * log_b(P). Therefore, log_(9)((3^2)^3) becomes 3 * log_(9)(3^2).Simplify Logarithm: Simplify the logarithm inside the expression. Since 9 is 3 squared, log_(9)(3^2) is asking ＂to what power do we raise 9 to get 3 squared?＂ The answer is 1 because 9 to the power of 1 is 9, and 3 squared is also 9. So, log_(9)(3^2) = 1.Multiply by Exponent: Multiply the simplified logarithm by the exponent. Now we have 3 * log_(9)(3^2) which simplifies to 3 * 1.Calculate Final Value: Calculate the final value. Multiplying 3 by 1 gives us the final answer of 3.

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$\log _{9} 729=$

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Algebra 2

Evaluate logarithms using properties

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\log _{9} 729=

Identify Relationship: Identify the relationship between the base of the logarithm and the number. $\newline$ The base of the logarithm is $9$ , and we need to find the exponent that $9$ must be raised to in order to get $729$ .
Express as Power: Express $729$ as a power of $9$ . We know that $9$ is $3$ squared ( $9 = 3^2$ ), and $729$ is $3$ to the sixth power ( $729 = 3^6$ ). Therefore, $729$ can be expressed as $(3^2)^3$ .
Rewrite Logarithm: Rewrite the logarithm using the new expression for $729$ . $\newline$ $\log_{9}(729)$ becomes $\log_{9}((3^2)^3)$ .
Apply Power Property: Apply the power property of logarithms. $\newline$ The power property states that $\log_b(P^Q) = Q \cdot \log_b(P)$ . Therefore, $\log_{9}((3^2)^3)$ becomes $3 \cdot \log_{9}(3^2)$ .
Simplify Logarithm: Simplify the logarithm inside the expression. $\newline$ Since $9$ is $3$ squared, $\log_{9}(3^2)$ is asking ＂to what power do we raise $9$ to get $3$ squared?＂ The answer is $1$ because $9$ to the power of $1$ is $9$ , and $3$ squared is also $9$ . So, $3$ $1$ .
Multiply by Exponent: Multiply the simplified logarithm by the exponent. $\newline$ Now we have $3 \times \log_{9}(3^2)$ which simplifies to $3 \times 1$ .
Calculate Final Value: Calculate the final value. $\newline$ Multiplying $3$ by $1$ gives us the final answer of $3$ .