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Evaluate. Write your answer as a whole number, proper fraction, or improper fraction in simplest form. $\newline$ $\log_3 81 =$ ______

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Evaluate logarithms

Full solution

Q. Evaluate. Write your answer as a whole number, proper fraction, or improper fraction in simplest form.

\newline

\log_3 81 =

______

Identify base and number: Identify the base of the logarithm and the number whose logarithm is to be found. $\newline$ In $\log_3 81$ , the base is $3$ and we need to find the power to which $3$ must be raised to get $81$ .
Rewrite as power of $3$ : Rewrite $81$ as a power of $3$ . $\newline$ $81$ can be expressed as $3 \times 3 \times 3 \times 3$ , which is equal to $3^4$ .
Substitute into logarithm: Substitute the expression for $81$ into the logarithm. $\newline$ $\log_3 81$ becomes $\log_3 3^4$ .
Apply logarithm power rule: Apply the logarithm power rule. $\newline$ The power rule states that $\log_b (a^c) = c \cdot \log_b a$ , where $b$ is the base of the logarithm, $a$ is the argument, and $c$ is the exponent. Since the base of the logarithm and the base of the exponent are the same ( $3$ in this case), $\log_3 3^4$ simplifies to just $4$ .

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