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Rewrite as a quotient of two common logarithms. Write your answer in simplest form. $\newline$ $\log_3 33 =$ ______

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Change of base formula

Full solution

Q. Rewrite as a quotient of two common logarithms. Write your answer in simplest form.

\newline

\log_3 33 =

______

Identify base and argument: Identify the base (b) and the argument (M) in the logarithmic expression $\log_3(33)$ . $\newline$ In this case, $b = 3$ and $M = 33$ .
Use change of base formula: Use the change of base formula to rewrite the logarithm in terms of common logarithms (base $10$ ). $\newline$ The change of base formula is $\log_b(M) = \frac{\log_c(M)}{\log_c(b)}$ , where $c$ is the new base, typically $10$ for common logarithms. $\newline$ Apply the formula: $\log_3(33) = \frac{\log(33)}{\log(3)}$ .
Apply the formula: Calculate the values of $\log(33)$ and $\log(3)$ using a calculator or logarithm table if necessary. $\newline$ However, for the purpose of this problem, we leave the expression in terms of $\log(33)$ and $\log(3)$ without numerical approximation.

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