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$Express the given expression as an integer or as a fraction in simplest form. log_(3)((1)/(3^(7))) Answer:$

Express the given expression as an integer or as a fraction in simplest form. $\newline$ $\log _{3}\left(\frac{1}{3^{7}}\right)$ $\newline$ Answer:

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Precalculus

Quotient property of logarithms

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Q. Express the given expression as an integer or as a fraction in simplest form.

\newline

\log _{3}\left(\frac{1}{3^{7}}\right)

\newline

Answer:

Understand the logarithm expression: Understand the logarithm expression $\log_{3}\left(\frac{1}{3^{7}}\right)$ . We need to express the logarithm of a fraction where the numerator is $1$ and the denominator is $3$ raised to the power of $7$ , with base $3$ .
Apply logarithm power rule: Apply the logarithm power rule. $\newline$ The power rule of logarithms states that $\log_b(a^c) = c \cdot \log_b(a)$ . In this case, we can apply the power rule to the denominator of the fraction. $\newline$ $\log_{3}\left(\frac{1}{3^{7}}\right) = \log_{3}(1) - \log_{3}(3^{7})$
Evaluate $\log_{3}(1)$ : Evaluate $\log_{3}(1)$ . $\newline$ The logarithm of $1$ to any base is always $0$ because any number raised to the power of $0$ is $1$ . $\newline$ $\log_{3}(1) = 0$
Apply power rule to $\log_{3}(3^7)$ : Apply the power rule to $\log_{3}(3^{7})$ . Using the power rule, we can take the exponent out in front of the logarithm. $\log_{3}(3^{7}) = 7 \times \log_{3}(3)$
Evaluate $\log_{3}(3)$ : Evaluate $\log_{3}(3)$ . The logarithm of a number to the same base is $1$ because any number raised to the power of $1$ is itself. $\log_{3}(3) = 1$
Combine results from Step $3$ and Step $5$ : Combine the results from Step $3$ and Step $5$ . $\newline$ Now we can combine the results to find the value of the original expression. $\newline$ $\log_{3}\left(\frac{1}{3^{7}}\right) = 0 - (7 \times 1)$
Simplify the expression: Simplify the expression. $\newline$ Subtracting $7$ times $1$ from $0$ gives us $-7$ . $\newline$ $\log_{3}\left(\frac{1}{3^{7}}\right) = 0 - 7 = -7$