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What is the value of
log_(9)((1)/(9)) ?
Answer:

Evaluate. $\newline$ $\log _{9} \frac{1}{9}$ $\newline$ Write your answer in simplest form.

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Full solution

Q. Evaluate.

\newline

\log _{9} \frac{1}{9}

\newline

Write your answer in simplest form.

Understand the problem: Understand the problem. $\newline$ We need to find the value of the logarithm of $\frac{1}{9}$ with base $9$ . The logarithm $\log_{\text{base}}(b)(a)$ answers the question: ＂To what power must we raise $b$ to obtain $a$ ?＂ In this case, we are looking for the power to which $9$ must be raised to get $\frac{1}{9}$ .
Apply the definition: Apply the definition of a logarithm. $\newline$ We know that $9$ to the power of $-1$ equals $\frac{1}{9}$ (since $\frac{1}{9}$ is the reciprocal of $9$ , and the reciprocal of a number is that number to the power of $-1$ ). Therefore, $\log_{9}\left(\frac{1}{9}\right)$ is asking for the exponent that makes $9$ equal to $\frac{1}{9}$ .
Calculate the logarithm: Calculate the logarithm. $\newline$ Since $9^{-1} = \frac{1}{9}$ , by the definition of logarithms, $\log_{9}\left(\frac{1}{9}\right) = -1$ .

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